Ideal Gases

State the meaning, in the equation, of the symbol $T$.

Identify the constant $B$.

State the meanings, in this equation, of the symbols $m$ and $\langle c^2 \rangle$.

$m$: ______
$\langle c^2 \rangle$: ______

Use the equations in (a) and (b) to derive an expression, in terms of $A$, $B$ and $T$, for the mean kinetic energy $E_K$ of a molecule of the gas.

$E_K$ = ______

On Fig. 4.1, sketch the variation with $T$ of the root-mean-square (r.m.s.) speed of the molecules of an ideal gas.

State what is meant by an ideal gas.

Show that the root-mean-square (r.m.s.) speed of molecules of this gas is approximately $500\ \text{m s}^{-1}$.

One molecule of the gas has a mass of $4.7 \times 10^{-26}\ \text{kg}$.

Determine the thermodynamic temperature of the gas.

temperature = ______ $\text{K}$

Calculate the internal energy $U$ of $6.0\ \text{mol}$ of the gas in (b). Explain your reasoning.

$U$ = ______ $\text{J}$

State the meaning of each of the symbols in this equation.

$p$: ______

$V$: ______

$N$: ______

$k$: ______

$T$: ______

Using the equation of state, derive an expression for the average translational kinetic energy $E_K$ of a particle in the gas in terms of some or all of $N$, $k$ and $T$.

$E_K$ = ______

A molecule of hydrogen gas consists of two hydrogen atoms, each of nucleon number 1. A molecule of oxygen gas consists of two oxygen atoms, each of nucleon number 16.

Assume that hydrogen and oxygen both behave as ideal gases.

A sample of hydrogen gas is at the same temperature as a sample of oxygen gas.

For the two samples, determine the ratio

ratio = ______

State what is meant by an ideal gas.

Use one of the basic assumptions of the kinetic theory to explain what can be deduced about the potential energy associated with the random motion of molecules in an ideal gas.

the number $N$ of molecules of the gas

$N$ = ______

the average translational kinetic energy $E_K$ of one molecule of the gas

$E_K$ = ______ $\text{J}$

the internal energy of the gas. Explain your reasoning.

internal energy = ______ $\text{J}$

The volume $V$ of the gas in (b) is now varied, keeping its pressure constant.

On Fig. 3.1, sketch the variation with $V$ of the internal energy $U$ of the gas.

State what is meant by an ideal gas.

Use one of the basic assumptions of the kinetic theory to explain what can be deduced about the potential energy associated with the random motion of molecules in an ideal gas.

the number $N$ of molecules of the gas

$N$ = ______

the average translational kinetic energy $E_K$ of one molecule of the gas

$E_K$ = ______ $\text{J}$

the internal energy of the gas. Explain your reasoning.

internal energy = ______ $\text{J}$

The volume $V$ of the gas in (b) is now varied, keeping its pressure constant.

On Fig. 3.1, sketch the variation with $V$ of the internal energy $U$ of the gas.

State what is meant by the Avogadro constant.

State the relationship between the Avogadro constant $N_A$, the molar gas constant $R$ and the Boltzmann constant $k$.

Complete Table 3.1 by giving expressions, in terms of some or all of $N$, $m$, $T$, $V$ and the constants in (a)(ii), for the quantities indicated.

Table 3.1

The temperature of sample X is now varied.

On Fig. 3.1, sketch the variation with thermodynamic temperature of the root-mean square (r.m.s.) speed of the molecules of the gas.

State three of the basic assumptions of the kinetic theory of gases.

Explain how molecular movement causes the pressure exerted by a gas.

Fig. 4.1 shows the variation with thermodynamic temperature $T$ of the mean-square speeds $\langle c^2 \rangle$ for two gases X and Y.

Fig. 4.2 shows the variation with $T$ of the product $pV$ for samples of the two gases, where $p$ is the pressure of the gas and $V$ is the volume of the gas.

State three conclusions about the gases and their samples that may be drawn from Fig. 4.1 and Fig. 4.2. The conclusions may be qualitative or quantitative. Use the space below for any working that you need.

State what is meant by the Avogadro constant.

State the relationship between the Avogadro constant $N_A$, the molar gas constant $R$ and the Boltzmann constant $k$.

Complete Table 3.1 by giving expressions, in terms of some or all of $N$, $m$, $T$, $V$ and the constants in (a)(ii), for the quantities indicated.

Table 3.1

The temperature of sample X is now varied.

On Fig. 3.1, sketch the variation with thermodynamic temperature of the root-mean square (r.m.s.) speed of the molecules of the gas.

State two of the basic assumptions of the kinetic theory of gases.

1 ______

2 ______

Determine an expression for the temperature $T$ of the gas in state X, in terms of $n$, $p$ and $V$.
Identify any other symbols that you use.

On Fig. 4.1, sketch the variation with volume of pressure for the gas as the gas undergoes the three changes. The state X is labelled. Label states Y and Z.

During the change of state from Y to Z, the increase in internal energy of the gas is $U$.
During the change of state from Z to X, the work done on the gas is $W$.

Complete Table 4.1 to indicate, for each of the three changes of state, the increase in internal energy of the gas, the thermal energy transferred to the gas and the work done on the gas, in terms of $p$, $V$, $U$ and $W$.

Table 4.1

State two of the basic assumptions of the kinetic theory of gases.

1 ______

2 ______

Determine an expression for the temperature $T$ of the gas in state X, in terms of $n$, $p$ and $V$.
Identify any other symbols that you use.

On Fig. 4.1, sketch the variation with volume of pressure for the gas as the gas undergoes the three changes. The state X is labelled. Label states Y and Z.

During the change of state from Y to Z, the increase in internal energy of the gas is $U$.
During the change of state from Z to X, the work done on the gas is $W$.

Complete Table 4.1 to indicate, for each of the three changes of state, the increase in internal energy of the gas, the thermal energy transferred to the gas and the work done on the gas, in terms of $p$, $V$, $U$ and $W$.

Table 4.1