An ideal gas at 17•c has a
pressure of 760 mmHg and is compressed
adiabatically until its volume is
halved.Calculate final temperature of the gas
assuming Cp= 2100 J/kg/K and Cv=1500J/kg/K
234
To calculate the final temperature of the gas after adiabatic compression, we can use the adiabatic equation:
P1 * V1^γ = P2 * V2^γ
Where:
P1 = Initial pressure of the gas
V1 = Initial volume of the gas
γ = Ratio of specific heats (Cp/Cv) for the gas
P2 = Final pressure of the gas
V2 = Final volume of the gas
In this case, we are given:
P1 = 760 mmHg
V1 = Initial volume
γ = Cp / Cv = 2100 J/kg/K / 1500 J/kg/K = 1.4
V2 = V1 / 2 = Initial volume / 2
We need to convert the initial pressure from mmHg to Pa, as well as the initial volume from whatever units it is given in to cubic meters (m^3) for consistency. Let's assume the initial volume is given in liters, so we'll need to convert it to cubic meters by dividing by 1000.
Now, we can start with the calculation.
Step 1: Convert the initial pressure from mmHg to Pa.
1 mmHg = 133.322 Pa
P1 = 760 mmHg * 133.322 Pa/mmHg
Step 2: Convert the initial volume to cubic meters.
Assuming the initial volume is given in liters, divide by 1000 to convert it to cubic meters.
V1 = Initial volume / 1000
Step 3: Substitute the values into the adiabatic equation.
P1 * V1^γ = P2 * V2^γ
Step 4: Rearrange the equation to solve for P2.
P2 = P1 * (V1 / V2)^γ
Step 5: Substitute the known values (P1, V1, γ) into the equation and calculate P2.
Step 6: Solve for the final temperature using the ideal gas law.
P2 * V2 = n * R * T2
Now, we can substitute the known values (P2, V2, R) into the equation, and solve for T2.
Note: In this explanation, I assumed that R (the gas constant) is already given or known. If it's not provided, you will need to find the appropriate value for R based on the units used for pressure, volume, and temperature.
Please provide the value for the initial volume of the gas to continue the calculation.