Heat at constant pressure (comparison)

Problem

The same $2\ \text{mol}$ of monatomic gas at $P_\text{atm} = 10^5\ \text{Pa}$ are heated from $T_A = 300\ \text{K}$ to $T_B = 350\ \text{K}$ with a free piston (constant pressure). How much heat is needed? Compare with the previous problem and explain the difference.

Solution

Molar heat at constant pressure. For a gas with $f$ degrees of freedom: $C_p = \frac{f+2}{2}\,R = \frac{5}{2}\,R \qquad (f = 3)$

Heat supplied. $Q = n\, C_p\, \Delta T = 2\cdot\frac{5}{2}\cdot 8{,}31\cdot 50 = 2077{,}5\ \text{J}$

$\ev{Q \approx 2{,}08\cdot 10^3\ \text{J}}$

Comparison with constant volume. In the previous problem $Q_V = 1246{,}5\ \text{J}$. The difference is: $Q - Q_V = n R\,\Delta T = 2\cdot 8{,}31\cdot 50 = 831\ \text{J}$ These $831\ \text{J}$ are the work done by the gas to push back the atmosphere during the constant-pressure expansion. That is why $C_p > C_V$: at constant pressure the heat must cover both the rise in internal energy and the work of expansion.

Links

Topics: Thermodynamics Concepts: First law of thermodynamics · Specific heat and heat capacity Objects: Piston and cylinder · Ideal gas