Suppose 2100 J of heat are added to 3.3 mol of argon gas at a constant pressure of 120 kPa. (Assume that the argon can be treated as an ideal monatomic gas.)
A) Find the change in internal energy.
B) Find the change in temperature for this gas.
C)Calculate the change in volume of the gas.
A) The change in internal energy of the gas can be found using the equation:
ΔU = q - W
where ΔU is the change in internal energy, q is the heat added to the system, and W is the work done by the system.
Since the process is at a constant pressure, the work done by the system is given by:
W = PΔV
where P is the constant pressure and ΔV is the change in volume.
Since the volume is not given, we need to find it first. We can use the ideal gas law equation:
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.
We can rearrange the equation to solve for V:
V = (nRT) / P
Plugging in the values,
V = (3.3 mol * 8.314 J/(mol*K) * T) / (120 kPa)
V = (3.3 mol * 8.314 J/(mol*K) * T) / (120,000 Pa)
Now we have the volume in terms of T, and we can substitute this into the equation for work done:
W = PΔV
W = (120 kPa) * [(3.3 mol * 8.314 J/(mol*K) * T) / (120,000 Pa)]
W = (120 * 10^3 N/m^2) * (3.3 mol * 8.314 J/(mol*K) * T) / (120 * 10^3 N/m^2)
W = 3.3 * 8.314 J/K * T
Now, we can substitute these values into the equation for ΔU:
ΔU = q - W
ΔU = 2100 J - (3.3 * 8.314 J/K * T)
B) To find the change in temperature, we can use the equation for heat:
q = nCΔT
where q is the heat added to the system, n is the number of moles, C is the molar heat capacity at constant pressure, and ΔT is the change in temperature.
Since argon is treated as an ideal monatomic gas, its molar heat capacity at constant pressure is given by:
C = (3/2)R
where R is the ideal gas constant.
Now we can substitute the values into the equation for heat:
2100 J = (3.3 mol) * ((3/2) * 8.314 J/(mol*K)) * ΔT
Simplifying,
2100 J = 3.3 * 3 * 8.314 J/K * ΔT
Solving for ΔT,
ΔT = 2100 J / (3.3 * 3 * 8.314 J/K)
C) To calculate the change in volume, we can use the equation:
ΔV = nRΔT / P
where ΔV is the change in volume, n is the number of moles, R is the ideal gas constant, ΔT is the change in temperature, and P is the pressure.
Now we can substitute the values into the equation:
ΔV = (3.3 mol * 8.314 J/(mol*K)) * ΔT / (120 kPa)
ΔV = (3.3 * 8.314 J/K) * ΔT / (120 * 10^3 N/m^2)
To find the answers to these questions, we need to use the first law of thermodynamics, which states that the change in internal energy (ΔU) of a system is equal to the heat (q) added to the system minus the work (w) done by the system:
ΔU = q - w
Since we are given that the heat added to the system is 2100 J and the process is carried out at constant pressure, we can use the equation:
q = ΔU + PΔV
where P is the pressure and ΔV is the change in volume.
A) Find the change in internal energy (ΔU):
We can rearrange the equation to solve for ΔU:
ΔU = q - PΔV
Substituting the given values, we have:
ΔU = 2100 J - (120 kPa)(ΔV)
Note that we need to convert the pressure to the appropriate unit for ΔV. In this case, we will use the ideal gas law to convert the pressure from kPa to Pa:
1 kPa = 1000 Pa
So, the pressure in Pa will be 120,000 Pa.
Now we can substitute the values into the equation:
ΔU = 2100 J - (120,000 Pa)(ΔV)
B) Find the change in temperature for this gas:
Since the process is carried out at constant pressure, we can use the equation for the heat capacity at constant pressure:
q = nCpΔT
where n is the number of moles, Cp is the molar heat capacity, and ΔT is the change in temperature.
We know that the heat added to the system (q) is 2100 J and the number of moles (n) is 3.3 mol. For monatomic gases, the molar heat capacity is given by:
Cp = (3/2)R
where R is the ideal gas constant. For argon, R = 8.314 J/(mol·K), so we can substitute these values into the equation:
2100 J = (3.3 mol)(3/2)(8.314 J/(mol·K))(ΔT)
Now we can solve for ΔT:
ΔT = (2100 J) / [(3.3 mol)(3/2)(8.314 J/(mol·K))]
C) Calculate the change in volume of the gas:
By rearranging the ΔU equation, we can solve for ΔV:
ΔV = (ΔU - 2100 J) / 120,000 Pa
Substitute the calculated ΔU value:
ΔU = 2100 J - (120,000 Pa)(ΔV) = (ΔU - 2100 J) / 120,000 Pa
Now we need to solve for ΔV.