Molar heat capacity at constant pressure of a diatomic gas is 3·5 R, where R is universal gas constant. The gas is heated by providing energy of 1400 J and is allowed to expand isobarically. Which one among the following represents the work done by the gas in this process ?
400 J
4900 J
560 J
280 J
Answer is Wrong!
Answer is Right!
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UPSC Geoscientist – 2024
Using Mayer’s relation, C_v = C_p – R = (7/2)R – R = (5/2)R.
The gas is heated isobarically, and energy Q = 1400 J is provided.
For an isobaric process, Q = nC_pΔT, where n is the number of moles and ΔT is the temperature change.
So, 1400 J = n * (7/2)R * ΔT.
By the First Law of Thermodynamics, Q = ΔU + W.
Work done by the gas W = Q – ΔU.
Change in internal energy for an ideal gas ΔU = nC_vΔT.
W = Q – nC_vΔT.
From the isobaric heat equation, nΔT = Q / C_p = 1400 / (7/2)R = 1400 * (2 / 7R) = 2800 / 7R = 400 / R.
Now substitute this into the work equation:
W = Q – nC_vΔT = 1400 – (nΔT) * C_v = 1400 – (400/R) * (5/2)R
W = 1400 – 400 * (5/2) = 1400 – 200 * 5 = 1400 – 1000 = 400 J.
Alternatively, W = nRΔT.
From Q = nC_pΔT, we have nΔT = Q/C_p.
W = R * (Q/C_p) = Q * (R/C_p).
Given C_p = (7/2)R, so R/C_p = R / ((7/2)R) = 2/7.
W = 1400 J * (2/7) = 200 * 2 = 400 J.