A gas has pressure p and volume V. It is now compressed adiabatically to 1/32 times the original volume. If (32)1.4= 128, the final pressure in terms of p isCorrect answer is '128'. Can you explain this answer?

Question

Answer

^{Adiabatic Process:}
An adiabatic process is a thermodynamic process in which there is no heat transfer to or from the system. This means that the system is insulated and does not exchange heat with its surroundings.

^{Given Information:}
- The initial pressure of the gas is p.
- The initial volume of the gas is V.
- The gas is compressed adiabatically to 1/32 times the original volume.
- The expression (32)1.4 evaluates to 128.

^Solution:
To find the final pressure of the gas in terms of p, we can use the adiabatic equation:

P1V1^γ = P2V2^γ

Where:
- P1 and V1 are the initial pressure and volume of the gas.
- P2 and V2 are the final pressure and volume of the gas.
- γ is the heat capacity ratio, which is equal to the ratio of specific heats (Cp/Cv) for the gas.

^{Step 1:} Identify the initial and final conditions.
- Initial conditions: P1 = p, V1 = V
- Final conditions: V2 = (1/32)V

^{Step 2:} Calculate the heat capacity ratio (γ).
The given expression (32)1.4 evaluates to 128, which means γ = 1.4.

^{Step 3:} Use the adiabatic equation to find the final pressure (P2).
P1V1^γ = P2V2^γ

Substituting the values:
p * V^1.4 = P2 * (1/32V)^1.4

Simplifying the equation:
p * V^1.4 = P2 * (1/32)^1.4 * V^1.4
p = P2 * (1/32)^1.4

Since (1/32)^1.4 is equal to 1/128, the equation becomes:
p = P2 * (1/128)

^{Step 4:} Solve for P2.
Multiply both sides of the equation by 128:
128p = P2

Therefore, the final pressure (P2) in terms of p is equal to 128.

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