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For an ideal monoatomic gas, molar heat ...

For an ideal monoatomic gas, molar heat capacity at constant volume `(C_(v))` is

A

`(2)/(3) R`

B

`(3)/(2) R`

C

`(5)/(2) R`

D

`(2)/(5) R`

Text Solution

Verified by Experts

The correct Answer is:
B
If a molecule of the ideal gas contains `n` atoms, then the number of degrees of freedom (different modes of translational, rotational, and and vibrational motion) is given by `3n` Three of these are assigned to the translational motion of the molecules, leaving `3n - 3` degrees of greedom for rotational and vibrational motion. For and ideal monoatomic gas, such as `He, 3 (1) - 3 = 0`, thus, the entire contribution to internal energy is from translational motion. Since each translation motion contributes `(1//2)RT` to the internal energy. we have for one mole of ideal monoatimic gas,
`U = 3 ((1)/(2) RT) = (3)/(2) RT`
`C_(V) = (Delta U)/(Delta T) = ((3)/(2) R Delta T)/(Delta T) = (3)/(2) R`
Since `C_(P) - C_(V) = R` (for 1 mol),
`C_(P) = R + C_(V) = R + (3)/(2) R = (5)/(2) R`
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