An ideal gas is sealed within a container at a temperature of 17°C and a pressure of 101 kPa. The container is heated until the temperature of the gas reaches100°C. A valve in the container is then opened to allow gas to escape until the pressure falls back to 101 kPa at 100°C.
a) Calculate the pressure in the gas just before the valve is opened.
b) Calculate the fraction of the initial mass of gas that was lost as a result of opening the valve
Pv/nT=R
Piv/nTi=pfv/nTf
To answer these questions, we can use the ideal gas law, which states:
PV = nRT
Where:
P - Pressure
V - Volume
n - Number of moles
R - Ideal gas constant
T - Temperature
In this scenario, we are given the initial and final temperatures, and the initial and final pressures. We can assume that the volume and the number of moles of gas remain constant.
a) To calculate the pressure just before the valve is opened, we can use the ideal gas law for the initial state of the gas.
PV = nRT
Solving for n, we get:
n = PV / RT
Since the number of moles remains constant, we can use this value to calculate the pressure just before the valve is opened.
b) To calculate the fraction of the initial mass of gas lost, we need to compare the number of moles of gas initially to the number of moles of gas remaining after the valve is opened.
Let's break down the steps to find the answers more clearly:
Step 1: Convert temperatures to Kelvin
To use the ideal gas law, we need to convert the temperatures from Celsius to Kelvin. The Kelvin temperature scale is obtained by adding 273.15 to the Celsius temperature.
Initial temperature, T1 = 17°C + 273.15 = 290.15 K
Final temperature, T2 = 100°C + 273.15 = 373.15 K
Step 2: Calculate the number of moles initially (n1)
Using the ideal gas law, we can calculate the number of moles initially.
n1 = (P1 * V) / (R * T1)
Step 3: Calculate the number of moles finally (n2)
We can calculate the number of moles finally using the ideal gas law.
n2 = (P2 * V) / (R * T2)
Step 4: Calculate the pressure just before the valve is opened (P3)
Since the volume and the number of moles remain constant, we can use the initial number of moles to find the pressure just before the valve is opened.
P3 = (n1 * R * T2) / V
Step 5: Calculate the fraction of the initial mass lost
The fraction of the initial mass lost can be calculated by comparing the initial and final number of moles:
Fraction lost = (n1 - n2) / n1
Now, let's calculate the answers using the given values:
P1 = 101 kPa
P2 = 101 kPa
V = (constant)
R = Ideal gas constant (8.31 J/(mol·K))
Step 1: Convert temperatures to Kelvin
T1 = 290.15 K
T2 = 373.15 K
Step 2: Calculate the number of moles initially (n1)
n1 = (P1 * V) / (R * T1)
Step 3: Calculate the number of moles finally (n2)
n2 = (P2 * V) / (R * T2)
Step 4: Calculate the pressure just before the valve is opened (P3)
P3 = (n1 * R * T2) / V
Step 5: Calculate the fraction of the initial mass lost
Fraction lost = (n1 - n2) / n1
By following these steps and plugging in the values, we can find both the pressure just before the valve is opened and the fraction of the initial mass lost.