Ideal-Gas Heat Capacities at Constant Volume and Pressure

All flashcards

What is $C_V$?

The molar heat capacity at constant volume: heat required per mole per kelvin when volume is fixed.

Why is expansion work zero in a rigid vessel?

The volume does not change, so $dV=0$ and $p\,dV=0$.

What does the first law become at constant volume?

With no expansion work, $dE_{\text{int}}=dQ$.

How are ideal-gas internal energy and temperature related?

For a fixed amount of ideal gas, $dE_{\text{int}}=nC_V\,dT$; internal energy depends only on temperature.

How does the number of active degrees of freedom affect $C_V$?

For $d$ active degrees of freedom, $C_V=\frac{d}{2}R$.

Quiz questions

1. For an ideal gas, what is the relation between the molar heat capacities at constant pressure and constant volume?Answer: $C_P=C_V+R$.
2. For an ideal gas heated at constant pressure, what is the expansion-work increment?Answer: $p\,dV=nR\,dT$.
3. Why is $C_P$ greater than $C_V$ for an ideal gas?Answer: At constant pressure, some added heat supplies expansion work.
4. An ideal gas has $d$ active molecular degrees of freedom. What is its molar heat capacity at constant pressure?Answer: $C_P=\left(\frac{d}{2}+1\right)R$.
5. For many dilute real gases, how does the measured difference $C_P-C_V$ compare with $R$?Answer: It is close to $R$.