Neon gas of a given mass expands isothermally to double volume. What should be the further fractional decrease in pressure, so that the gas when adiabatically compressed from that state reaches the original state?

Question

Answer

Given Information:

Mass of Neon gas

Expansion of gas to double volume

Question:

Fractional decrease in pressure required for adiabatic compression to reach original state

Solution:

Let us assume that the initial pressure and volume of the gas to be P1 and V1 respectively. The final pressure and volume are P2 and V2 respectively. The process of expansion is isothermal, so the temperature remains constant during the process. Hence, we can apply the ideal gas law to find the initial pressure of the gas:

P1V1 = nRT

where n is the number of moles of neon gas, R is the universal gas constant, and T is the temperature of the gas.

Since the temperature remains constant during the isothermal expansion, we can write:

P1V1 = P2V2

At this stage, the volume of the gas is doubled. Hence, V2 = 2V1. Substituting this value in the above equation, we get:

P1 = 2P2

Now, let us assume that the gas is adiabatically compressed to reach the original state. During adiabatic compression, the temperature of the gas increases since no heat is exchanged with the surroundings. The relationship between pressure and volume during adiabatic compression is given by:

PV^γ = constant

where γ is the ratio of specific heats of the gas.

Since the process is adiabatic, we can write:

P1V1^γ = P2V2^γ

Substituting the values of P1, V1, P2, and V2, we get:

2P2(2V1)^γ = P2V1^γ

Simplifying the above equation, we get:

P2/P1 = 2^(1/(γ-1))

Therefore, the fractional decrease in pressure required for adiabatic compression to reach the original state is:

(P1 - P2)/P1 = (P1 - P1/2)/P1 = 1/2

Conclusion:

The fractional decrease in pressure required for adiabatic compression to reach the original state is 1/2.

Answer

10/3

Answer

10/3 only

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