The Molar Specific Heats of an Ideal Gas — Question 3

Question 3

Mass-specific and molar heat capacities encode different comparisons

Use $pV=nRT$, $R=8.314$ J/(mol K), and the sign convention $\Delta U=Q-W$, with $W$ the work done by the gas. Only pressure–volume work occurs. Gas A has molar mass .00400 kg/mol and $C_V=\tfrac32R$; gas B has molar mass .0280 kg/mol and $C_V=\tfrac52R$. Separate 10.0 g samples receive 100 J each at fixed volume.

1. Calculate each mass-specific heat $c_V=C_V/M$ and temperature rise.

2. Find the mass of B with the same total heat capacity as the 10.0 g A sample. Explain why comparing only molar capacities gives the wrong equal-mass heating prediction.

Question 3 - Solution

Mass-specific and molar heat capacities encode different comparisons

(a) Method and calculation

$$c_{V,A}=\frac{1.5(8.314)}{.00400}=3117.75\,\mathrm{J/(kg\,K)},\quad c_{V,B}=\frac{2.5(8.314)}{.0280}\approx742.321\,\mathrm{J/(kg\,K)}.$$$$\Delta T_A=\frac{100}{.0100(3117.75)}\approx3.20744\,\mathrm K,\quad\Delta T_B=\frac{100}{.0100(742.321)}\approx13.4713\,\mathrm K.$$

(b) Interpretation and further calculation

$$m_Bc_{V,B}=m_Ac_{V,A}\ \Rightarrow\ m_B=.0100\frac{(1.5/.00400)}{(2.5/.0280)}=.0420\,\mathrm{kg}.$$$$\frac{c_{V,A}}{c_{V,B}}=4.20.$$Equal masses contain different mole counts; the lighter species can store more thermal energy per kilogram despite its smaller molar capacity.

Physical check

Heat capacity in J/K, molar capacity in J/(mol K) and mass-specific capacity in J/(kg K) must not be interchanged.