The molar specific heat of an ideal gas at constant pressure is `C_(p) = 5/2 R`. Some amount of this gas in a closed container has a volume `0.0083 m^(3)`, a temperature 300 K and a pressure `1.6 xx 10^(6) N cdot m^(-2)`. If `2.49 xx 10^(4) J`1 of heat is supplied at a constant volume of the gas, find out the final temperature and pressure. Given, `R = 8.3 J cdot mol^(-1) cdot K^(-1)`.

Explanation

Let n mol of the gas be present in the container.
`:. pV = nRT`
or, `n = (pV)/(RT)`
`=((1.6 xx 10^6)xx0.0083)/(8.3 xx 300) = 5.33`
Now, `C_(v) = C_(p) - R = 5/2 R - R = 3/2 R`
`= 1.5 xx 83 J cdot mol^(-1) cdot K^(-1)`
Heat supplied at constant volume,
`Q_(v) = nC_(v)(T_f - T_i)`
or, `T_(f) - T_i = (Q_v)/(nC_v) = (2.49 xx 10^(4))/(5.33 xx (1.5 xx 8.3)) = 375.2 K`
or, `T_(f) = T_(i) + 375.2 = 300 + 375.2 = 675.2 K`
`:.` Final temperature is 675.2 K.
or, `p_(f) = p_(i)(T_f)/(T_i) = (1.6 xx 10^(6)) xx (675.2)/(300)`
`= 3.6 xx 10^(6) N cdot m^(-2)`.
`:.` Final pressure is `3.6 xx 10^(6) N cdot m^(-2)`.