# Which Equation Represents the Combined Gas Law

**aviationbrief.com** – Which Equation Represents the Combined Gas Law

### Learning Objectives

By the finish of this section, you will be able to:

- Country the ideal gas law in terms of molecules and in terms of moles.
- Utilize the platonic gas law to summate pressure modify, temperature modify, volume change, or the number of molecules or moles in a given volume.
- Use Avogadro’s number to convert betwixt number of molecules and number of moles.

In this department, we go along to explore the thermal behavior of gases. In particular, we examine the characteristics of atoms and molecules that compose gases. (About gases, for example nitrogen, N_{ii}, and oxygen, O_{2}, are equanimous of two or more atoms. We will primarily use the term “molecule” in discussing a gas considering the term can likewise exist applied to monatomic gases, such as helium.)

Gases are hands compressed. We tin come across prove of this in Tabular array 1 in Thermal Expansion of Solids and Liquids, where you will notation that gases take the

*largest*

coefficients of volume expansion. The large coefficients hateful that gases expand and contract very apace with temperature changes. In addition, you will note that most gases expand at the

*same*

rate, or take the aforementioned

*β*. This raises the question every bit to why gases should all act in almost the aforementioned way, when liquids and solids have widely varying expansion rates.

The answer lies in the large separation of atoms and molecules in gases, compared to their sizes, every bit illustrated in Figure ii. Considering atoms and molecules accept large separations, forces betwixt them tin be ignored, except when they collide with each other during collisions. The motion of atoms and molecules (at temperatures well in a higher place the humid temperature) is fast, such that the gas occupies all of the accessible volume and the expansion of gases is rapid. In dissimilarity, in liquids and solids, atoms and molecules are closer together and are quite sensitive to the forces between them.

To get some idea of how pressure level, temperature, and volume of a gas are related to one some other, consider what happens when you pump air into an initially deflated tire. The tire’s volume first increases in directly proportion to the corporeality of air injected, without much increment in the tire pressure. Once the tire has expanded to well-nigh its full size, the walls limit volume expansion. If we go on to pump air into it, the pressure increases. The force per unit area volition further increase when the car is driven and the tires move. Most manufacturers specify optimal tire force per unit area for cold tires. (Run across Figure 3.)

At room temperatures, collisions between atoms and molecules can be ignored. In this case, the gas is chosen an platonic gas, in which example the human relationship betwixt the pressure level, volume, and temperature is given by the equation of country chosen the ideal gas law.

### Ideal Gas Law

The

*ideal gas law*

states that*PV*

=*NkT*, where

*P*

is the accented pressure of a gas,

*Five*

is the volume it occupies,

*N*

is the number of atoms and molecules in the gas, and

*T*

is its absolute temperature. The constant

*k*

is chosen the

*Boltzmann constant*

in honor of Austrian physicist Ludwig Boltzmann (1844–1906) and has the value*k*

= 1.38 × ten^{−23}

J/Thou.

The ideal gas law can be derived from bones principles, but was originally deduced from experimental measurements of Charles’ constabulary (that volume occupied by a gas is proportional to temperature at a fixed pressure) and from Boyle’s law (that for a fixed temperature, the product

*PV* is a constant). In the platonic gas model, the volume occupied by its atoms and molecules is a negligible fraction of

*Five*. The ideal gas law describes the beliefs of real gases nether most weather. (Annotation, for case, that

*N*

is the total number of atoms and molecules, independent of the type of gas.)

Let the states come across how the ideal gas police force is consistent with the beliefs of filling the tire when information technology is pumped slowly and the temperature is abiding. At first, the pressure

*P*

is substantially equal to atmospheric pressure, and the volume

*Five*

increases in straight proportion to the number of atoms and molecules

*N*

put into the tire. Once the volume of the tire is constant, the equation

*PV*

=*NkT* predicts that the pressure should increase in proportion to

*the number N of atoms and molecules*.

### Example 1. Calculating Pressure Changes Due to Temperature Changes: Tire Pressure level

Suppose your bicycle tire is fully inflated, with an absolute pressure of 7.00 × ten^{5}

Pa (a estimate pressure of just under 90.0 lb/in^{2}) at a temperature of xviii.0ºC. What is the force per unit area after its temperature has risen to 35.0ºC? Presume that there are no appreciable leaks or changes in book.

#### Strategy

The pressure in the tire is irresolute only because of changes in temperature. Commencement we need to identify what nosotros know and what we want to know, then identify an equation to solve for the unknown.

We know the initial pressure level

*P*

_{}

= 7.00 × 10^{5}

Pa, the initial temperature

*T*

_{} = 18.0ºC, and the final temperature

*T*

_{f} = 35.0ºC. Nosotros must notice the last pressure level

*P*

_{f}. How can we use the equation

*PV*

=

*NkT*? At beginning, it may seem that not enough information is given, because the volume

*V*

and number of atoms

*N*

are not specified. What we can do is employ the equation twice:

*P*

_{}

*V*

_{} = NkT_{}

and

*P*

_{f}

*5*

_{f} = NkT_{f}. If nosotros divide

*P*

_{f}

*V*

_{f}

by

*P*

_{}

*V*

_{}

we can come up with an equation that allows us to solve for

*P*

_{f}.

[latex]\displaystyle\frac{P_{\text{f}}V_{\text{f}}}{P_0V_0}=\frac{N_{\text{f}}kT_{\text{f}}}{N_0kT_0}\\[/latex]

Since the volume is constant,

*5*

_{f}

and

*V*

_{}

are the aforementioned and they cancel out. The same is true for

*N*

_{f}

and

*N*

_{}, and

*1000*, which is a constant. Therefore,

[latex]\displaystyle\frac{P_{\text{f}}}{P_0}=\frac{T_{\text{f}}}{T_0}\\[/latex]

We can then rearrange this to solve for

*P*

_{f}: [latex]P_{\text{f}}=P_0\frac{T_{\text{f}}}{T_0}\\[/latex], where the temperature must be in units of kelvins, because

*T*

_{}

and

*T*

_{f}

are absolute temperatures.

#### Solution

Catechumen temperatures from Celsius to Kelvin:

*T*

_{}

= (18.0 + 273)K = 291 K

*T*

_{f}

= (35.0 + 273)K = 308 One thousand

Substitute the known values into the equation.

[latex]\displaystyle{P}_{\text{f}}=P_0\frac{T_{\text{f}}}{T_0}=7.00\times10^five\text{ Pa}\left(\frac{308\text{ K}}{291\text{ Grand}}\right)=7.41\times10^5\text{ Pa}\\[/latex]

#### Discussion

The concluding temperature is about six% greater than the original temperature, so the final force per unit area is about six% greater too. Annotation that

*absolute*

pressure and

*absolute*

temperature must exist used in the ideal gas police.

### Making Connections: Take-Home Experiment—Refrigerating a Balloon

Inflate a balloon at room temperature. Go out the inflated balloon in the fridge overnight. What happens to the balloon, and why?

### Example two. Calculating the Number of Molecules in a Cubic Meter of Gas

How many molecules are in a typical object, such as gas in a tire or water in a drink? Nosotros tin can use the ideal gas law to give us an idea of how large

*N*

typically is.

Calculate the number of molecules in a cubic meter of gas at standard temperature and pressure (STP), which is defined to be 0ºC and atmospheric pressure.

#### Strategy

Because pressure, book, and temperature are all specified, we tin can use the ideal gas law

*PV*

=*NkT*, to find

*N*.

#### Solution

Identify the knowns:

[latex]\begin{array}{lll}T&=&0^{\circ}\text{C}=273\text{ One thousand}\\P&=&1.01\times10^5\text{ Pa}\\V&=&1.00\text{ m}^3\\thou&=&1.38\times10^{-23}\text{ J/K}\terminate{array}\\[/latex]

Identify the unknown: number of molecules,

*N*.

Rearrange the ideal gas law to solve for

*Northward*:

[latex]\brainstorm{array}{lll}PV&=&NkT\\N&=&\frac{PV}{kT}\stop{array}\\[/latex]

Substitute the known values into the equation and solve for

*N*:

[latex]\displaystyle{N}=\frac{PV}{kT}=\frac{\left(one.01\times10^v\text{ Pa}\right)\left(1.00\text{ k}^three\right)}{\left(ane.38\times10^{-23}\text{ J/1000}\correct)\left(273\text{ K}\right)}=two.68\times10^{25}\text{ molecules}\\[/latex]

#### Discussion

This number is undeniably big, because that a gas is generally empty infinite.

*Due north*

is huge, even in small volumes. For example, ane cm^{3} of a gas at STP has 2.68 × 10^{19}

molecules in information technology. One time again, notation that

*N*

is the same for all types or mixtures of gases.

## Moles and Avogadro’southward Number

It is sometimes convenient to work with a unit other than molecules when measuring the amount of substance. A

*mole*

(abbreviated mol) is defined to be the amount of a substance that contains as many atoms or molecules as there are atoms in exactly 12 grams (0.012 kg) of carbon-12. The actual number of atoms or molecules in one mole is called

*Avogadro’s number*

(*N*

_{A}), in recognition of Italian scientist Amedeo Avogadro (1776–1856). He developed the concept of the mole, based on the hypothesis that equal volumes of gas, at the same pressure and temperature, incorporate equal numbers of molecules. That is, the number is independent of the type of gas. This hypothesis has been confirmed, and the value of Avogadro’s number is*N*

_{A}

= 6.02 × x^{23}

mol^{−1}.

### Avogadro’due south Number

I mole always contains 6.02 × x^{23}

particles (atoms or molecules), contained of the element or substance. A mole of whatsoever substance has a mass in grams equal to its molecular mass, which can be calculated from the atomic masses given in the periodic table of elements.

*N*

_{A}

= six.02 × 10^{23}

mol^{−one}

### Bank check Your Understanding

The active ingredient in a Tylenol pill is 325 mg of acetaminophen (C_{8}H_{ix}NO_{two}). Observe the number of active molecules of acetaminophen in a single pill.

#### Solution

Nosotros first need to summate the molar mass (the mass of one mole) of acetaminophen. To do this, we demand to multiply the number of atoms of each element by the element’s atomic mass.

(8 moles of carbon)(12 grams/mole) + (9 moles hydrogen)(ane gram/mole) + (1 mole nitrogen)(14 grams/mole) + (2 moles oxygen)(16 grams/mole) = 151 g

And then we need to summate the number of moles in 325 mg.

[latex]\displaystyle\left(\frac{325\text{ mg}}{151\text{ grams/mole}}\right)\left(\frac{i\text{ gram}}{1000\text{ mg}}\correct)=2.xv\times10^{-iii}\text{ moles}\\[/latex]

Then use Avogadro’s number to calculate the number of molecules.

*Due north*

= (ii.15 × 10^{−3}

moles)(half dozen.02 × 10^{23}

molecules/mole) = 1.thirty × x^{21}

molecules

### Example 3. Computing Moles per Cubic Meter and Liters per Mole

Calculate the following:

- The number of moles in 1.00 yard
^{iii}of gas at STP - The number of liters of gas per mole.

#### Strategy and Solution

- We are asked to notice the number of moles per cubic meter, and we know from Example two that the number of molecules per cubic meter at STP is ii.68 × x
^{25}. The number of moles can exist plant by dividing the number of molecules by Avogadro’s number. We permit

*n*

correspond the number of moles,

[latex]{g}\text{ mol/yard}^iii=\frac{N\text{ molecules/m}^3}{half-dozen.02\times10^{23}\text{ molecules/mol}}=\frac{two.68\times10^{25}\text{ molecules/chiliad}^3}{6.02\times10^{23}\text{ molecules/mol}}=44.5\text{ mol/1000}^iii\\[/latex] - Using the value obtained for the number of moles in a cubic meter, and converting cubic meters to liters, we obtain [latex]\frac{\left(ten^3\text{ L/one thousand}^3\correct)}{44.five\text{ mol/m}^3}=22.5\text{ 50/mol}\\[/latex]

#### Discussion

This value is very close to the accepted value of 22.4 50/mol. The slight deviation is due to rounding errors caused by using iii-digit input. Once more this number is the same for all gases. In other words, information technology is independent of the gas.

The (average) molar weight of air (approximately 80% North_{2}

and xx% O_{2}

is

*M*

= 28.eight g. Thus the mass of 1 cubic meter of air is ane.28 kg. If a living room has dimensions 5 m × 5 m × 3 m, the mass of air inside the room is 96 kg, which is the typical mass of a human.

### Bank check Your Understanding

The density of air at standard conditions (*P*

= 1 atm and*T*= 20ºC) is 1.28 kg/thou^{iii}. At what pressure level is the density 0.64 kg/yard^{three}

if the temperature and number of molecules are kept constant?

#### Solution

The best style to approach this question is to think about what is happening. If the density drops to half its original value and no molecules are lost, then the volume must double. If we look at the equation

*PV*

=*NkT*, we run across that when the temperature is constant, the pressure is inversely proportional to volume. Therefore, if the volume doubles, the pressure must drop to half its original value, and

*P*

_{f}

= 0.50 atm.

## The Ideal Gas Law Restated Using Moles

A very common expression of the ideal gas law uses the number of moles,

*northward*, rather than the number of atoms and molecules,

*N*. We offset from the ideal gas law,*PV*

=*NkT*, and multiply and divide the equation past Avogadro’south number

*N*

_{A}. This gives [latex]PV=\frac{N}{N_{\text{A}}}N_{\text{A}}kT\\[/latex].

Note that [latex]n=\frac{Northward}{N_{\text{A}}}\\[/latex] is the number of moles. We define the universal gas abiding

*R*=*N*

_{A}

*yard*, and obtain the ideal gas law in terms of moles.

### Platonic Gas Law (in terms of moles)

The ideal gas law (in terms of moles) is*PV*

=*nRT*.

The numerical value of

*R*

in SI units is*R*

=*N*

_{A}

*k*

= (half-dozen.02 × ten^{23}

mol^{−1})(one.38 × 10^{−23}

J/Thousand) = 8.31 J/mol · Yard.

In other units,

*R*

= 1.99 cal/mol · K

*R*

= 0.0821 L · atm/mol · K

You can apply whichever value of

*R*

is most convenient for a detail trouble.

### Case 4. Calculating Number of Moles: Gas in a Bike Tire

How many moles of gas are in a cycle tire with a volume of 2.00 × 10^{−three}

k^{3}(2.00 Fifty), a pressure level of 7.00 × 10^{v}

Pa (a gauge force per unit area of simply nether 90.0 lb/in^{2}), and at a temperature of eighteen.0ºC?

#### Strategy

Place the knowns and unknowns, and cull an equation to solve for the unknown. In this case, we solve the ideal gas law,

*PV*

=*nRT*, for the number of moles

*due north*.

#### Solution

Identify the knowns:

[latex]\brainstorm{array}{lll}P&=&vii.00\times10^5\text{ Pa}\\V&=&ii.00\times10^{-3}\text{ g}^3\\T&=&eighteen.0^{\circ}\text{C}=291\text{ K}\\R&=&8.31\text{ J/mol}\cdot\text{ K}\cease{array}\\[/latex]

Rearrange the equation to solve for

*n*

and substitute known values.

[latex]\brainstorm{array}{lll}n&=&\frac{PV}{RT}=\frac{\left(7.00\times10^5\text{ Pa}\right)\left(2.00\times10^{-3}\text{ thou}^3\right)}{\left(viii.31\text{ J/mol}\cdot\text{ K}\right)\left(291\text{ K}\right)}\\\text{ }&=&0.579\text{ mol}\finish{array}\\[/latex]

**Discussion**

The most convenient selection for

*R*

in this example is 8.31 J/mol · Chiliad, because our known quantities are in SI units. The pressure and temperature are obtained from the initial conditions in Case i, but we would become the same answer if we used the terminal values.

The ideal gas police can exist considered to be another manifestation of the constabulary of conservation of energy (see Conservation of Energy). Work done on a gas results in an increase in its free energy, increasing pressure and/or temperature, or decreasing book. This increased free energy can also be viewed as increased internal kinetic energy, given the gas’s atoms and molecules.

## The Ideal Gas Constabulary and Energy

Let united states now examine the role of energy in the behavior of gases. When yous inflate a bike tire by hand, you exercise work by repeatedly exerting a force through a altitude. This energy goes into increasing the force per unit area of air within the tire and increasing the temperature of the pump and the air.

The ideal gas law is closely related to energy: the units on both sides are joules. The right-hand side of the platonic gas law in

*PV*

=*NkT* is*NkT*. This term is roughly the amount of translational kinetic energy of

*N*

atoms or molecules at an absolute temperature

*T*, equally we shall encounter formally in Kinetic Theory: Atomic and Molecular Caption of Pressure and Temperature. The left-hand side of the ideal gas law is

*PV*, which as well has the units of joules. We know from our report of fluids that pressure is one type of potential energy per unit volume, then pressure multiplied by volume is energy. The important point is that in that location is energy in a gas related to both its force per unit area and its volume. The energy tin be changed when the gas is doing work as it expands—something we explore in Heat and Heat Transfer Methods—like to what occurs in gasoline or steam engines and turbines.

### Problem-Solving Strategy: The Platonic Gas Law

**Footstep 1.**

Examine the situation to determine that an ideal gas is involved. About gases are nearly ideal.

**Step 2.** Make a list of what quantities are given, or tin can be inferred from the problem as stated (identify the known quantities). Convert known values into proper SI units (K for temperature, Pa for pressure, 1000^{3}

for volume, molecules for

*N*, and moles for

*due north*).

**Step 3.**

Identify exactly what needs to exist determined in the trouble (identify the unknown quantities). A written list is useful.

**Step iv.**

Decide whether the number of molecules or the number of moles is known, in order to determine which course of the platonic gas police force to apply. The showtime form is

*PV*

=*NkT* and involves

*N*, the number of atoms or molecules. The second form is

*PV*=*nRT* and involves

*n*, the number of moles.

**Step 5.**

Solve the platonic gas police for the quantity to be adamant (the unknown quantity). You may need to have a ratio of final states to initial states to eliminate the unknown quantities that are kept fixed.

**Step 6.**

Substitute the known quantities, along with their units, into the appropriate equation, and obtain numerical solutions complete with units. Be sure to use absolute temperature and absolute force per unit area.

**Step 7.**

Bank check the answer to meet if it is reasonable: Does it make sense?

### Check Your Understanding

Liquids and solids have densities near k times greater than gases. Explain how this implies that the distances between atoms and molecules in gases are near 10 times greater than the size of their atoms and molecules.

#### Solution

Atoms and molecules are close together in solids and liquids. In gases they are separated by empty infinite. Thus gases have lower densities than liquids and solids. Density is mass per unit of measurement volume, and volume is related to the size of a body (such as a sphere) cubed. So if the altitude between atoms and molecules increases past a cistron of 10, and so the volume occupied increases by a factor of thousand, and the density decreases by a factor of g.

## Department Summary

- The ideal gas law relates the pressure and book of a gas to the number of gas molecules and the temperature of the gas.
- The ideal gas police can be written in terms of the number of molecules of gas:
*PV*=

*NkT*, where

*P*

is pressure level,

*Five*

is volume,

*T*

is temperature,

*N*

is number of molecules, and 1000 is the Boltzmann constant*k*= 1.38 × 10^{–23}

J/Grand. - A mole is the number of atoms in a 12-yard sample of carbon-12.
- The number of molecules in a mole is called Avogadro’southward number

*NA*,*NA*= vi.02 × 10^{23}

mol^{−1}. - A mole of any substance has a mass in grams equal to its molecular weight, which can be determined from the periodic table of elements.
- The ideal gas law tin also exist written and solved in terms of the number of moles of gas:
*PV*=

*nRT*, where north is number of moles and

*R*

is the universal gas constant,*R*

= 8.31 J/mol ⋅ K. - The ideal gas law is by and large valid at temperatures well above the boiling temperature.

### Conceptual Questions

Discover out the homo population of Earth. Is in that location a mole of people inhabiting Earth? If the average mass of a person is lx kg, calculate the mass of a mole of people. How does the mass of a mole of people compare with the mass of World?

Under what circumstances would you expect a gas to deport significantly differently than predicted by the ideal gas police force?

A constant-volume gas thermometer contains a fixed corporeality of gas. What property of the gas is measured to indicate its temperature?

### Bug & Exercises

- The gauge force per unit area in your car tires is two.50 × ten
^{5}

Due north/m^{ii}

at a temperature of 35.0ºC when you drive it onto a ferry gunkhole to Alaska. What is their gauge force per unit area later on, when their temperature has dropped to –40.0ºC? - Convert an absolute pressure level of 7.00 × 10
^{v}

N/m^{2}

to gauge pressure in lb/in^{ii}. (This value was stated to be just less than 90.0 lb/in^{2}

in Example four. Is it?) - Suppose a gas-filled incandescent lite bulb is manufactured so that the gas within the bulb is at atmospheric pressure when the bulb has a temperature of 20.0ºC. (a) Find the gauge pressure inside such a bulb when information technology is hot, bold its average temperature is 60.0ºC (an approximation) and neglecting whatsoever alter in book due to thermal expansion or gas leaks. (b) The actual terminal pressure for the low-cal bulb volition be less than calculated in part (a) because the drinking glass bulb will expand. What will the actual terminal pressure be, taking this into account? Is this a negligible difference?
- Large helium-filled balloons are used to lift scientific equipment to loftier altitudes. (a) What is the pressure inside such a balloon if it starts out at sea level with a temperature of 10.0ºC and rises to an altitude where its volume is 20 times the original book and its temperature is –50.0ºC? (b) What is the gauge force per unit area? (Presume atmospheric force per unit area is constant.)
- Confirm that the units of nRT are those of free energy for each value of R: (a) 8.31 J/mol ⋅ K, (b) 1.99 cal/mol ⋅ Thou, and (c) 0.0821 Fifty ⋅ atm/mol ⋅ K.
- In the text, it was shown that

*Due north*/*V*= 2.68 × x^{25}

one thousand^{−3}

for gas at STP. (a) Show that this quantity is equivalent to

*N*/*Five*

= two.68 × ten^{19}

cm^{−three}, every bit stated. (b) About how many atoms are at that place in one μm^{3}

(a cubic micrometer) at STP? (c) What does your answer to part (b) imply about the separation of atoms and molecules? - Calculate the number of moles in the 2.00-50 volume of air in the lungs of the average person. Note that the air is at 37.0ºC (body temperature).
- An airplane passenger has 100 cm
^{3}

of air in his tummy just before the plane takes off from a sea-level airdrome. What volume will the air have at cruising distance if cabin pressure drops to seven.50 × x^{4}

Northward/m^{two}? - (a) What is the book (in km
^{3}) of Avogadro’s number of sand grains if each grain is a cube and has sides that are one.0 mm long? (b) How many kilometers of beaches in length would this cover if the beach averages 100 g in width and 10.0 m in depth? Neglect air spaces between grains. - An expensive vacuum organization tin can achieve a force per unit area as depression equally ane.00 × 10
^{–seven}

N/grand^{2}

at 20ºC. How many atoms are there in a cubic centimeter at this force per unit area and temperature? - The number density of gas atoms at a certain location in the space above our planet is well-nigh one.00 × 10
^{eleven}

m^{−three}, and the pressure is two.75 × 10^{–10}

N/m^{ii}

in this space. What is the temperature there? - A bicycle tire has a pressure of vii.00 × x
^{5}

N/chiliad^{2}

at a temperature of xviii.0ºC and contains 2.00 L of gas. What will its pressure be if you let out an amount of air that has a volume of 100cm3 at atmospheric pressure? Presume tire temperature and volume remain abiding. - A high-pressure gas cylinder contains l.0 L of toxic gas at a pressure of i.forty × 10
^{seven}

Northward/m^{2}

and a temperature of 25.0ºC. Its valve leaks after the cylinder is dropped. The cylinder is cooled to dry ice temperature (–78.5ºC) to reduce the leak charge per unit and pressure so that it tin be safely repaired. (a) What is the terminal force per unit area in the tank, assuming a negligible amount of gas leaks while being cooled and that there is no phase change? (b) What is the final pressure if i-10th of the gas escapes? (c) To what temperature must the tank be cooled to reduce the pressure to 1.00 atm (bold the gas does not change phase and that at that place is no leakage during cooling)? (d) Does cooling the tank announced to be a practical solution? - Find the number of moles in 2.00 50 of gas at 35.0ºC and nether 7.41 × 10
^{vii}

N/m^{ii}

of force per unit area. - Calculate the depth to which Avogadro’s number of table tennis assurance would cover World. Each brawl has a diameter of 3.75 cm. Assume the space between assurance adds an actress 25.0% to their volume and assume they are not crushed by their own weight.
- (a) What is the judge pressure in a 25.0ºC car tire containing 3.sixty mol of gas in a xxx.0 L volume? (b) What will its estimate pressure exist if you add one.00 L of gas originally at atmospheric force per unit area and 25.0ºC? Assume the temperature returns to 25.0ºC and the volume remains constant.
- (a) In the deep infinite between galaxies, the density of atoms is as low as x
^{six}

atoms/m^{three}, and the temperature is a frigid 2.seven K. What is the pressure? (b) What volume (in thou^{3}) is occupied by one mol of gas? (c) If this volume is a cube, what is the length of its sides in kilometers?

## Glossary

**ideal gas law:** the physical law that relates the force per unit area and book of a gas to the number of gas molecules or number of moles of gas and the temperature of the gas

**Boltzmann constant:**

*one thousand*, a concrete constant that relates free energy to temperature;

*k*= ane.38 × 10^{–23}

J/Chiliad

**Avogadro’due south number:**

*NA*, the number of molecules or atoms in one mole of a substance;

*NA*

= 6.02 × 10^{23}

particles/mole

**mole:** the quantity of a substance whose mass (in grams) is equal to its molecular mass

### Selected Solutions to Problems & Exercises

one. 1.62 atm

3. (a) 0.136 atm; (b) 0.135 atm. The deviation between this value and the value from part (a) is negligible.

5. (a) [latex]\text{nRT}=\left(\text{mol}\right)\left(\text{J/mol}\cdot \text{Yard}\right)\left(\text{K}\right)=\text{J}\\[/latex];

(b) [latex]\text{nRT}=\left(\text{mol}\right)\left(\text{cal/mol}\cdot \text{K}\right)\left(\text{K}\correct)=\text{cal}\\[/latex];

(c) [latex]\begin{assortment}{lll}\text{nRT}& =& \left(\text{mol}\right)\left(\text{L}\cdot \text{atm/mol}\cdot \text{Grand}\right)\left(\text{Thousand}\right)\\ & =& \text{L}\cdot \text{atm}=\left({\text{1000}}^{3}\right)\left({\text{North/m}}^{two}\right)\\ & =& \text{N}\cdot \text{g}=\text{J}\cease{assortment}\\[/latex]

7. seven.86 × 10^{−2}

mol

ix. (a) 6.02 × 10^{five}

km^{iii}; (b) six.02 × 10^{8}

km

eleven. −73.9ºC

13. (a) 9.14 × ten^{6}

N/yard^{2}; (b) 8.23 × 10^{half dozen}

N/yard^{two}; (c) 2.16 K; (d) No. The final temperature needed is much likewise depression to exist hands achieved for a large object.

15. 41 km

17. (a) 3.7 × 10^{−17}

Pa; (b) 6.0 × 10^{17}

thou^{three}; (c) 8.4 × 10^{ii}

km

### Which Equation Represents the Combined Gas Law

Source: https://courses.lumenlearning.com/suny-physics/chapter/13-3-the-ideal-gas-law/