Question

Derive the following statement

"T(temperatue)-V(volume) and P(pressure)-V(volume) relationship in the adiabatic changes"

Answer 1

Equation of state for an ideal gas is

PV = nRT …………..1

where P is gas pressure, V is volume, n is the number of moles, R is the universal gas constant (8.314) and T is the absolute temperature.

Now we will write the first law of thermodynamics, the conservation of energy, in differential form as

dq = du + p*dV…………………….2

where dq is a thermal energy given to the gas, du is a change in the internal energy of the gas, and p*dV is the work done by the gas while expanding through the change in volume dV.

As the gas has a specific heat at constant volume of C_V, then we may set dq = nC_V dT. So,

du = nC_V dT…………………………3

Adiabatic process is the process where there is no exchange of matter and heat from outside the system.

Therefore q = constant and dq = 0

from 2 and 3, 0 = nC_V dT + p*dV i.e., internal energy of the gas may be reduced in favor of expansion, or vice versa. This expression can be written in an equivalent form as

0 = (C_V/R)(dT/T) + dV/V………………….4

(lets divide first term nRT, and the second term by pV).

Now, from 1,

P*dV + V*dp = nRdT

or

dp/p + dV/V = dT/T…………………….5

(after divided the Left Hand Side by pV, and the Right Hand Side by nRT).

Now we will be using equation 5 in 4

0 = (C_V/R)(dp/p + dV/V) + dV/V

-(C_V/R) dp/p = (1 + C_V/R) dV/V.

Cp = Cv + R

C_V/R = C_V/(C_p - C_V) = 1/(n-1), where n is Cp/Cv

-dp/p = n dV/V

Integrating…

P0/P = (V/V0)^n

i.e., the pressure varies inversely as the volume raised to the power n.

To calculation of V, equation to write dV/V = dT/T - dp/p, and substitute for dV/V in eq. 4

(C_V/R + 1) dT/T = dp/p

Proceeding as before produces the result

p/p0 = (T/T0) ^(n/n-1)

i.e the pressure varies directly as the temperature reaised to power n