Question

A unit mass of carbon dioxide gas is contained in a cylinder behind a piston and is heated from p1 = 552 kPa and V1 = 0.0184 m3
to p2 = 138 kPa. It expands acc
ording to a law pV1.3 = constant and the heat supplied to the gas during
the expansion process is 10.125 kJ. Calculate, per unit of mass:
(a) the work done during the of expansion process.
(b) the change in the internal energy of carbon dioxide.

Answer 1

Solution. Find the final value of the volume during expansion. According to the condition of the problem

$$p_1V^{1.3}_1=p_2V^{1.3}_2$$

therefore

$$V_2=V_1(\frac{p_1}{p_2})^{\frac{1}{1.3}}=0.0184(\frac{552\times 10^3}{138\times 10^3})^{\frac{1}{1.3}}\approx 0.0534 m^3$$

a) Find the work done during the of expansion process is equal to 

$$W_{12}=\int_1^2\frac{p_1V^{1.3}_1}{V^{1.3}}dV=\frac{p_2V_2-p_1V_1}{1-1.3}$$

$$W_{12}==\frac{138\times 10^3 \times 0.0534-552\times 10^3 \times 0.0184}{-0.3}=9292J$$

b) From the first law of thermodynamics

$$\Delta U=Q-W$$

where ∆U is change in internal energy; Q=10.125kJ is heat added to the system; W is work done by system.

As result get

$$\Delta U=10125-9292=833J$$

Answer. a) 9292J b) 833J.