A cylinder contains 10 moles of an ideal gas at a temperature of 300 K. The gas is compressed at constant pressure until the final volume equals 0.77 times the initial volume. The molar heat capacity at constant volume of the gas is 24.0 J/mol ∙ K and R = 8.31 J/mol ∙ K . How much heat is absorbed by the gas during this process? (R = 8.31 J/mol ∙ K)
A) -22 kJ
B) -17 kJ
C) 22 kJ
D) 17 kJ
E) -5.7 kJ

Question

Quizplus · Accepted Answer

Since the process is taking place at constant pressure, we can use the equation Q = nCpΔT, where n is the number of moles, Cp is the molar heat capacity at constant pressure, and ΔT is the change in temperature. However, we do not know ΔT, so we must use the ideal gas law to find it.

PV = nRT

Initial: P1V1 = nRT1
Final: P1(0.77V1) = nRT2

Dividing the two equations:

(0.77V1 / V1) = (T2 / T1)

T2 = (0.77T1)

Now we can use this to find ΔT:

ΔT = T2 - T1 
ΔT = 0.77T1 - T1 
ΔT = -0.23T1

Plugging this into the equation for Q:

Q = nCpΔT 
Q = (10 mol)(24.0 J/mol K)(-0.23T1) 
Q = -55.2T1

Finally, we can use the initial ideal gas law equation to find T1:

P1V1 = nRT1

P1 = nRT1 / V1

P1 = (10 mol)(8.31 J/mol K)(300 K) / V1

P1 = 249.3 / V1

Now we can substitute this into the equation for Q:

Q = -55.2(249.3 / V1) 
Q = -13717.6 / V1

Q = -13717.6 / (0.77V1)

Q = -17811.7 J or -17.8 kJ (rounded to two significant figures)

Therefore, the answer is (A) -22 kJ.

A Cylinder Contains 10 Moles of an Ideal Gas at a Temperature