an ideal monatomic gas is allow to expand slowly until its pressure is reduced to exactly half its original value. By what factor those the volume changes if the pressure is (1)isothermal (2)adiabatic.

Please is the answer suppose to be a calculation or an explanation

The answer is a calculation, which I have already performed. This is the third post of this question.

Please the answer

The answer to this question requires a calculation as well as an explanation.

(1) For an isothermal process, the temperature of the gas remains constant. In this case, we can use the ideal gas law, which states that PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the ideal gas constant, and T is the temperature.

Since the gas is monatomic, it has a unique gas constant, given by R = (3/2) * R₀, where R₀ is the universal gas constant.

For an isothermal process, the temperature is constant, so we can write the equation as:
P₁V₁ = P₂V₂

We want to find the factor by which the volume changes, so we can express it as:
V₂/V₁ = P₁/P₂

Given that P₂ equals half of P₁, we have:
V₂/V₁ = P₁/(P₁/2) = 2

Therefore, for an isothermal process, the volume changes by a factor of 2.

(2) For an adiabatic process, there is no heat exchange with the surroundings, meaning the temperature can change. For an adiabatic process, we can use the equation:
P₁V₁^γ = P₂V₂^γ

Here, γ is the heat capacity ratio, which is equal to 5/3 for a monatomic gas.

We want to find the factor by which the volume changes, so we can express it as:
(V₂/V₁)^γ = P₁/P₂

Given that P₂ equals half of P₁, we have:
(V₂/V₁)^γ = P₁/(P₁/2) = 2

To solve for (V₂/V₁), we take the γth root of both sides:
V₂/V₁ = 2^(1/γ)

Substituting γ = 5/3, we have:
V₂/V₁ = 2^(1/(5/3)) = 2^(3/5) = 1.515

Therefore, for an adiabatic process, the volume changes by a factor of approximately 1.515.

In summary, the volume changes by a factor of 2 for an isothermal process and by a factor of approximately 1.515 for an adiabatic process.