The temperature of 2.10 mol of an ideal monatomic gas is raised 17.0 K in an adiabatic process. What are (a) the work W done by the gas, (b) the energy transferred as heat Q, (c) the change ΔEint in internal energy of the gas, and (d) the change ΔK in the average kinetic energy per atom?
To solve this problem, we can use the equations for adiabatic processes in an ideal gas:
(a) The work done by the gas can be calculated using the formula:
W = -nCvΔT
Where:
- W is the work done by the gas
- n is the number of moles of the gas (2.10 mol)
- Cv is the molar heat capacity at constant volume (for a monatomic ideal gas, Cv = (3/2)R, where R is the ideal gas constant)
- ΔT is the change in temperature (17.0 K)
Substituting the values into the equation, we get:
W = -2.10 mol * (3/2)R * 17.0 K
(b) Since the process is adiabatic (no heat transfer), Q = 0.
(c) The change in internal energy can be calculated using the first law of thermodynamics:
ΔEint = Q - W
Since Q = 0, we have:
ΔEint = -W
Substituting the previously calculated value for W, we get:
ΔEint = 2.10 mol * (3/2)R * 17.0 K
(d) The change in average kinetic energy per atom can be calculated by dividing the change in internal energy by the number of particles (atoms) in the gas. The number of atoms can be calculated using Avogadro's number, which is approximately 6.022 x 10^23 atoms per mole.
ΔK = ΔEint / (n * Avogadro's number)
Substituting the previously calculated values, we get:
ΔK = (2.10 mol * (3/2)R * 17.0 K) / (2.10 mol * 6.022 x 10^23 atoms/mol)
Simplifying the expression, we have:
ΔK = (3/2)R * 17.0 K / (6.022 x 10^23)
Now, we can calculate the values of (a), (b), (c), and (d) using the given information and equations:
(a) The work done by the gas:
W = -2.10 mol * (3/2)R * 17.0 K
(b) The energy transferred as heat:
Q = 0
(c) The change in internal energy of the gas:
ΔEint = 2.10 mol * (3/2)R * 17.0 K
(d) The change in average kinetic energy per atom:
ΔK = (3/2)R * 17.0 K / (6.022 x 10^23)
To find the values of work done by the gas (W), energy transferred as heat (Q), change in internal energy (ΔEint), and change in average kinetic energy per atom (ΔK), we need to use the equations for adiabatic processes and ideal gases.
(a) Work done by the gas (W) can be calculated using the equation:
W = -ΔEint
Since it's an adiabatic process, no heat is transferred, meaning Q = 0. Therefore, W = -ΔEint = 0.
(b) Energy transferred as heat (Q) in an adiabatic process is equal to zero, as mentioned above. So, Q = 0.
(c) The change in internal energy (ΔEint) of the gas can be calculated using the equation:
ΔEint = (3/2) n R ΔT
where n is the number of moles of the gas, R is the ideal gas constant, and ΔT is the change in temperature.
Given:
Number of moles of the gas (n) = 2.10 mol
Change in temperature (ΔT) = 17.0 K
Using these values:
ΔEint = (3/2) (2.10 mol) R (17.0 K)
R is the ideal gas constant, which has a value of 8.314 J/(mol K). Plugging in this value:
ΔEint = (3/2) (2.10 mol) (8.314 J/(mol K)) (17.0 K)
Simplifying the equation gives us the value of ΔEint.
(d) The change in average kinetic energy per atom (ΔK) can be found using the equation:
ΔK = (3/2) R ΔT
Given the same values for ΔT and R as before:
ΔK = (3/2) (8.314 J/(mol K)) (17.0 K)
Simplifying the equation gives us the value of ΔK.