Consider one mole of perfect gas in a cylinder of unit cross-section with a piston attached. A spring is attached to the piston and to the bottom of the cylinder. Initially the spring is unstretched and the gas is in equilibrium. A certain amount of heat Q is supplied to the gas causing an increase of volume from V0 to V1.

a) what is the initial pressure of the system?

b) what is the final pressure of the system?

c) using the first law of thermodynamics, write down the relation between Q, Pa, V, V0, and k.

Answers (1)

 

  1. Piston is considered massless and balanced by atmospheric pressure. So the initial 

pressure of system inside cylinder is Pa

  1. On supplying heat volume of gas increases from V_{0}to V_1


Increase in Volume=V_1-V_{0}  =Area of base×height=A\times x

V_{1}-V_{0}=A\times x

x=\frac{V_{1}-V_{0}}{A}

(Force exerted by spring)

F=Kx=\frac{K(V_1-V_0)}{A}

(Force due to spring on unit area)

 F=K(V_{1}-V_0)

(Final total pressure on gas)

 P_f=P_a+K(V_1-V_0)

  1. dQ=dU+dW (First law of thermodynamics)

dU=C_v(T-T_0)

T=final temperature of the gas

T0=initial temperature of the gas

n=1 

T_y=T=\frac{P_fV_f}{R}=\frac{\left [P_a+K(V_1-V_0) \right ]}{R}

dW=P_a(V_1-V_0)+\frac{1}{2}kx^{2}(Increase in potential energy of spring)

dQ=dU+dW=C_v(T-T_0)+P_a(V_1-V_0)+\frac{1}{2}kx^2

dQ=dU+dW=C_v(T-T_0)+P_a(V_1-V_0)+\frac{1}{2}k(V_1-V_0)^2.

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