Question

16g of nitrogen gas at STP are pressurized in an isochoric process to a pressure of 25atm .

A-What is the final temperature?

B-What is the work done on the gas? W=0 I got that , it's the correct answer

C-What is the heat input to the gas?

D-What is the pressure ratio pmax/ pmin ?

please answer this problem not simaler problems. thank u

Answer 1

An ideal gas undergoing an adiabatic, reversible process satisfies the condition:
p?V^? = constant
throughout the process.

? = Cp/Cv is the heat capacity ratio. For a diatomic ideal gas like nitrogen it is
? = (7/2)?R / (5/2)?R = 7/5

Using ideal gas you may rewrite adiabatic equation as:
p^(?-1)?T^(-?) = constant

Hence:
p?^(?-1) / T?^? = p?^(?-1) / T?^?
=>
T? = T? ? (p?/p?)^[(?-1)/?]
= 273.15K ? (25atm / 1atm)^[(7/5-1)/(7/5)]
= 273.15K ? (25)^[2/7]
= 684.81K

The work done on the gas is given by the integral
W = - ? p dV from V? to V?

V can expressed in terms of V by adiabatic equation:
p?V^? = constant = p??V?^?
=>
V = V??p?^(1/?) ?p^(-1/?)
=>
dV = V??p?^(1/?) ? (-1/?)?p^(-1/? - 1) dp
=>
W = - ? p dV from V? to V?
= - ? p ? V??p?^(1/?) ? (-1/?)?p^(-1/? - 1) dp from p? to p?
= V??p?^(1/?) ? (1/?) ? ? p^(-1/?) dp from p? to p?
= V??p?^(1/?) ? (1/?) ? [1/(1 - 1/?)] ? { p?^[(1 - 1/?)] - p?^[(1 - 1/?)] }
= V??p?^(1/?) ? [1/(? - 1)] ? { p?^[(1 - 1/?)] - p?^[(1 - 1/?)] }
= V??p? ? [1/(? - 1)] ? { (p?/p?)^[(1 - 1/?)] - 1 }
= n?R?T? ? [1/(? - 1)] ? { (p?/p?)^[(1 - 1/?)] - 1 }

For this process
W = (16g/28g/mol) ? 8.314472J/molK ? 273.15K ? (5/2) ? { (25)^(2/7) -1}
= 4893.88 J