A monoatomic ideal gas is initially at temperature T_o, pressure p_o, and volume
ID: 3162355 • Letter: A
Question
A monoatomic ideal gas is initially at temperature T_o, pressure p_o, and volume V_o, (state a in the figure). The pressure in the gas is increased isochorically to a value 4p_o (state b). The gas then expands isothermally to a volume 2V_o (state c) and finally returns to its original state following a straight line on a p-V diagram (see figure). What are the values of volume, temperature, and pressure is states b and c? Remember to write your answers in terms of only the input parameters T_o, p_o, and V_o. What are the values of work W, change in internal energy Delta U, and heat transfer Q in process a-b? What are the values of work W, change in internal energy Delta U, and heat transfer Q in process b-c? What are the values of work W, change in internal energy Delta U, and heat transfer Q in process c-a? What is the efficiency of an engine using this cycle?Explanation / Answer
a) isochoric process means constant volume. For monoatomic ideal gas equation of state is PV=nRT
That means Cv=3/2
now at point B P0V0/T0= P1V1/T1 where P1=4P0 and V0=V1 then
P0V0/T0=4P0V0/T1 or T1=4T0
at the point C temperature remain constant
then P1V1/T1=P2V2/T2 where T1=T2= 4T0, V2= 2V0
then P1*4V0/T1=P2*2V0/T1
or P2=8P0
Therefore P1=4P0,V1=V0, T1= 4T0
and P2=8P0,V2=2V0,T2=4T0
b)the a-b process is isochoric that means dv=0 that means internal energy=total workdone =heat trasfer
therefore internal energy/heat transfer= mCv*change in temperature= m*3/2*(4T0-T0)=3/2m*3T0=9/2mT0
c) In b-c process we get isothermal situation that means internal energy= heat transfer=nRTln{V2/V0}=n*8.314*4T0ln(2V0/V0)=23.051nT0
d)In process c-a the volume compressed from 2V0 to V0 so its an adiabatic compression.
Internal energy U=nRT0 and work done W= K(V2^1--V0^1-)/1- where for ideal monoatomic gas= 1.66
using values W=K(2V0)^1-1.66- V0^1-1.66/1-1.66= -k/.66(2V0)^-.66-V0^-.66
e) The efficiency of the cycle= workdone/heat absorbed= 9/2mT0+23.051nT0/ -k/.66(2V0)^-.66-V0^-.66