Question

Suppose that 4.8 moles of an ideal diatomic gas has a temperature of 1061 K, and that each molecule has a mass 2.32 × 10-26 kg and can move in three dimensions, rotate in two dimensions, and vibrate in one dimension as the bond between the atoms stretches and compresses. It may help you to recall that the number of gas molecules is equal to Avagadros number (6.022 × 1023) times the number of moles of the gas.

a) How many degrees of freedom does each molecule of the gas have?

b) What is the internal energy of the gas?

c) What is the average translational speed of the gas molecules?

d) The gas cools to a temperature 524 K, which causes the gas atoms to stop vibrating, but maintain their translational and rotational modes of motion. What is the change in the internal energy of the gas?

Answer 1

(a) three dimensions, rotate in two dimensions, and vibrate in one dimension

means

degree of freedom = 6

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(b)

U = (N/2) * n * T

where N is degree of freedom

U = (6/2) * 4.8 * 8.314 * 1061

U = 127024.62 J

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(c)

v = sqrt ( 3RT / m)

where m = 2.32e-26 * 6.022e23 = 0.01397

so,

v = sqrt ( 3 * 8.314 * 1061 / 0.01397)

v = 1376.28 m/s

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(d)

now, when T = 524 K

One degree of freedom will decrease as it stops vibrating, so we have only 5 degree of freedom

U ( new) = (5/2) * 4.8 * 8.314 * 524

U = 52278.4 J

so,

change in the internal energy of the gas = 52278.4 J - 127024.62

change in the internal energy of the gas = - 74746.2 J