N moles of an ideal diatomic gas are in a cylinder at temperature T. suppose on supplying heat to the gas, its temperature remain constant but n moles get dissociated into atoms. Heat supplied to the gas is

Options

A. Zero
B. `(1)/(2)nRT`
C. `(3)/(2)nRT`
D. `(3)/(2) (N-n)RT`

Answer

B

Explanation

Since the gas in enclosed in a vessel. therefore during heating process, volume of the gas remain constant. Hence no work is done by the gas. It means heat supplied to the gas is used to increase its ionternal energy only.
Initial energy of the gas is
`U_(1)=N((5)/(2)R)R`
Since n mole get dissociated into atom, therefore after heating, vessel contain (N-n) mole of diatomic gas 2n moles of a mnoatomic gas.
Hence the internal energy for the gas, after heating will be equal to
`U_(2)=(N-n) ((5)/(2)R)T+2n((3)/(2)R)T`
Hence, the heat supplied = increase in internal energy
`U_(2)-U_(2) =(1)/(2)nRT `
Alternate :
`U_(i) =N((5)/(2)R)T`
`U_(f)=(N-n) ((5)/(2)R)T`
` U_(f)=(N-n) ((5)/(2)R) T+2n((3)/(2)R)T`
W = 0 (Closed system )
So, `q=Delta U`
`q=U_(f) -U_(i)=(1)/(2)nRT`