Question

The ratio of the molar heat capacities of an ideal gas is $C_p/C_v=7/6.$ Calculate the change in internal energy of 1.0 mole of the gas when its temperature is raised by 50 K .

(a) Keeping the pressure constant

(b) Keeping the volume constant and

(c) Adiabatically.

Answer 1

$\left({\frac{C_p}{C}}_v\right)=\frac{7}{6},n=1mol,\Delta T=50K$

(a) Keeping the pressure constant,

dQ = dU + dW

$\Delta T=50K,y=\frac{7}{6},$ n = 1 mol.

dq = Du + Dw

$\Rightarrow n{C}_{p}dT=dU+RdT$

$\Rightarrow dU=n{C}_{p}dT-RdT$

= 1 $\times \frac{Ry}{(y-1)}\times dT-RdT$

= 7 RdT - RdT

= 7 RdT-RdT = 6 RdT

= $6\times 8.3\times 50=2490J.$

(b) Keeping volume constant,

dU = $n{C}_v$ dT

= $1\times \frac{R}{y-1}\times dT$

= $1(\frac{8.3}{\frac{7}{6}}-1)\times 50$

= $8.3\times 50\times 6$ = 2490 J.

(c) Adiabatically, dQ - 0,

dU = -dW

= $\left[\frac{n\times R}{y-1}\right(T_1-T_2\left)\right]$

= $\frac{1\times 8.3}{\frac{7}{6}-1}=(T_2-T_3)$
= $8.3\times 6\times 50=2490J.$