Question

An ideal monatomic gas is allowed to expand adiabatically so that its pressure and temperature related as P = CT*x then x is equal to (C is a constant)​

Answer 1

the value of x = 2.5

An ideal monatomic gas is allowed to expand adiabatically so that its pressure and temperature related as $\bf{P=CT^x}$.....(1) , where C is constant.

we know for adiabatic process, $\bf{PV^{\gamma}=constant}$

from gas law, PV = nRT

so, V = nRT/P

then, $P\left(\frac{nRT}{P}\right)^{\gamma}=k$

⇒ $P^{(1-\gamma)}T^{\gamma}=\frac{k}{(nR)^{\gamma}}$

let k/(nR)^y = C(another constant)

⇒$P=CT^{\frac{\gamma}{\gamma-1}}$.....(2)

on comparing equations (1) and (2),

$\frac{\gamma}{\gamma-1}=x$

for monoatomic gas, $\gamma=\frac{5}{3}$

so, x = (5/3)/(5/3-1) = (5/3)/(2/3) = 2.5

hence, value of x = 2.5

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Answer 2

$\huge\bold\purple{Answer:-}$

An ideal monatomic gas is allowed to expand adiabatically so that its pressure and temperature related as P = CT*x then x is equal to (C is a constant)