A quantity of monatomic ideal gas expands adiabatically from a volume of 2.0 liters to 6.0 liters. If the initial pressure is P0, what is the final pressure?
In an expansion of this type,
P*V^g = constant
where g is the specific heat ratio Cp/Cv, which for a monatomic gas is 5/3. Let Pf be the final pressure. The final volume is Vf = 3 Vo. Thus
Po*Vo^5/3 = Pf*(3Vo)^5/3
Pf/Po = (1/3)^(5/3) = 0.160
Well, let's see. An ideal gas expanding adiabatically...sounds like it's really on a roll! 🌬️✨ Now, to find the final pressure, we can use the formula for adiabatic expansion, which is P0V0^(γ) = P1V1^(γ), where P0 is the initial pressure, V0 is the initial volume, P1 is the final pressure, and V1 is the final volume. The γ stands for the heat capacity ratio of the gas.
So, in this case, we have P0(2.0 liters)^(γ) = P1(6.0 liters)^(γ). Since we don't know the specific heat capacity ratio, we can't calculate the final pressure just yet. However, we can make some assumptions based on typical values. For monatomic gases, the heat capacity ratio (γ) is usually around 5/3, so we can use that as an estimate.
With that in mind, we can write the equation as P0(2.0 liters)^(5/3) = P1(6.0 liters)^(5/3). Solving for P1, we get P1 ≈ P0(2.0 liters/6.0 liters)^(5/3). Simplifying further, we find that P1 ≈ P0/2.
So, the final pressure is approximately half of the initial pressure. But remember, this is just an estimate based on the assumption of a heat capacity ratio for monatomic gases. It would be best to check the specific heat capacity ratio for the gas you're dealing with to get a more accurate answer.
To find the final pressure of the gas, we can use the adiabatic expansion formula:
P1 * V1^γ = P2 * V2^γ
Where:
P1 = initial pressure
V1 = initial volume
P2 = final pressure (what we want to find)
V2 = final volume
γ = heat capacity ratio (for monatomic ideal gas, γ = 5/3)
Given values:
V1 = 2.0 liters
V2 = 6.0 liters
P1 = P0 (initial pressure)
Substituting the known values into the formula, we have:
P0 * (2.0)^γ = P2 * (6.0)^γ
To find P2, we can rearrange the formula:
P2 = P0 * (2.0/6.0)^γ
Substituting the value of γ = 5/3:
P2 = P0 * (2.0/6.0)^(5/3)
Simplifying the expression:
P2 = P0 * (1/3)^(5/3)
Therefore, the final pressure (P2) of the gas is P0 multiplied by (1/3) raised to the power of (5/3).
To find the final pressure of the gas, we can use the adiabatic expansion formula, which relates the initial and final pressure, volume, and the ratio of specific heat capacities (γ) for a monatomic ideal gas:
P1 * V1^γ = P2 * V2^γ
Where:
P1 = initial pressure
V1 = initial volume
P2 = final pressure (what we need to find)
V2 = final volume
γ = ratio of specific heat capacities (γ = 5/3 for a monatomic ideal gas)
Given values:
P1 = P0 (initial pressure)
V1 = 2.0 liters (initial volume)
V2 = 6.0 liters (final volume)
γ = 5/3
Now we can rearrange the equation to solve for P2:
P2 = P1 * (V1/V2)^γ
Substituting the given values:
P2 = P0 * (2.0/6.0)^(5/3)
Simplifying:
P2 ≈ P0 * 0.395
Therefore, the final pressure (P2) is approximately 0.395 times the initial pressure (P0).