Consider a cylinder containing 2 mol of an ideal gas at an initial temperature of 25 C. The cylinder is fitted with a piston on which a weight is placed, subjecting the gas to a constant pressure of 2.66 atm. The cylidner is then wrapped with a heater, from which 1.50 kJ of heat is absorbed by the gas, forcing the piston to move upwards under constant pressure.
What is the final temperature of the gas?
To determine the final temperature of the gas, we need to use the equation:
q = nCΔT
where:
q is the heat absorbed by the gas,
n is the number of moles of the gas,
C is the molar heat capacity of the gas,
and ΔT is the change in temperature.
In this case, we are given:
q = 1.50 kJ = 1.50 × 10^3 J (converting from kJ to J)
n = 2 mol
ΔT = ? (what we need to find)
To calculate the molar heat capacity of an ideal gas at constant pressure, we can use the equation:
Cp = 5/2 R
where Cp is the molar heat capacity at constant pressure and R is the gas constant (8.314 J/mol·K).
Plugging in the values, we have:
Cp = 5/2 × 8.314 J/mol·K
Cp ≈ 20.785 J/mol·K
Now, we can rearrange the equation for heat to solve for the change in temperature as follows:
q = nCpΔT
Rearranging the equation, we have:
ΔT = q / (nCp)
Substituting the given values:
ΔT = (1.50 × 10^3 J) / (2 mol × 20.785 J/mol·K)
Calculating:
ΔT ≈ 36.10 K
To find the final temperature, we add the change in temperature to the initial temperature:
Final temperature = Initial temperature + ΔT
Final temperature = 25°C + 36.10 K
Converting the final temperature back to Celsius:
Final temperature ≈ 98.1°C
Therefore, the final temperature of the gas is approximately 98.1°C.
To find the final temperature of the gas, you can use the ideal gas law equation:
PV = nRT
Where:
P = Pressure (in atm)
V = Volume (in L)
n = Number of moles of gas
R = Ideal gas constant (0.0821 L.atm/mol.K)
T = Temperature (in Kelvin)
First, let's convert the given temperature from Celsius to Kelvin:
Initial temperature = 25 °C = 298 K
Next, we need to find the initial volume of the gas, but it is not provided in the question. We can assume, for simplicity, that the volume doesn't change during the process. Therefore, the initial volume (V1) is not needed to find the final temperature.
Now, we can rearrange the ideal gas law equation to solve for the final temperature (T2):
T2 = (P2 * V1) / (n * R)
Since the pressure (P1 = 2.66 atm) remains constant, we can rewrite the equation as:
T2 = (P1 * V1) / (n * R)
Using the given pressure (P1 = 2.66 atm), number of moles (n = 2 mol), and the ideal gas constant (R = 0.0821 L.atm/mol.K), we can substitute the values into the equation:
T2 = (2.66 atm * V1) / (2 mol * 0.0821 L.atm/mol.K)
Now, we can focus on the heat absorbed by the gas, which is 1.50 kJ. The heat absorbed by the gas is given by the equation:
q = nCΔT
Where:
q = Heat absorbed (in J)
n = Number of moles of gas
C = Molar heat capacity (in J/mol.K)
ΔT = Change in temperature (in K)
Since the pressure is constant during the process, we can simplify the equation to:
q = nCΔT
And rearrange it to solve for ΔT:
ΔT = q / (n * C)
We know the heat absorbed by the gas (q = 1.50 kJ = 1500 J), the number of moles (n = 2 mol), and the molar heat capacity (C). However, the molar heat capacity is not provided in the question, so it cannot be determined with the given information.
To find the final temperature, we need to know the molar heat capacity.