Thermodynamics

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58 of 190

Question 58 of 190

A certain gas is isothermally compressed to (13)rd\left(\dfrac{1}{3}\right)^{rd} of its initial volume (Vo=3V_o = 3 litre) by applying required pressure. If the bulk modulus of the gas is 3×1053 \times 10^5 N/m2^2, the magnitude of work done on the gas is _______ J.

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Answer: 989

For an isothermal process, the isothermal bulk modulus equals the gas pressure:
BT=−VdPdVB_T = -V\dfrac{dP}{dV}

Since PV=constantPV = \text{constant}, differentiating gives P dV+V dP=0P\,dV + V\,dP = 0, so dPdV=−PV\dfrac{dP}{dV} = -\dfrac{P}{V}, which yields BT=PB_T = P.

Therefore, the initial pressure is:
Po=3×105 N/m2P_o = 3 \times 10^5\text{ N/m}^2

Volumes:
Vo=3 L=3×10−3 m3V_o = 3\text{ L} = 3 \times 10^{-3}\text{ m}^3
Vf=13Vo=10−3 m3V_f = \dfrac{1}{3}V_o = 10^{-3}\text{ m}^3

Work done on the gas during an isothermal compression:
Won=−Wby=−PoVoln⁡(VfVo)=PoVoln⁡(VoVf)W_{\text{on}} = -W_{\text{by}} = -P_o V_o \ln\left(\dfrac{V_f}{V_o}\right) = P_o V_o \ln\left(\dfrac{V_o}{V_f}\right)

Won=(3×105)(3×10−3)ln⁡3W_{\text{on}} = (3 \times 10^5)(3 \times 10^{-3})\ln 3

Won=900ln⁡3 JW_{\text{on}} = 900 \ln 3\text{ J}

Using ln⁡3≈1.0986\ln 3 \approx 1.0986:
Won≈900×1.0986≈988.74 JW_{\text{on}} \approx 900 \times 1.0986 \approx 988.74\text{ J}

Rounded to the nearest integer:
Won≈989 JW_{\text{on}} \approx 989\text{ J}

Hence, the work done on the gas is approximately 989 J989\text{ J}.

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