The temperature of 4.60 mol of an ideal monatomic gas is raised 15.0 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 find the values of (a) work done by the gas, (b) energy transferred as heat, (c) change in internal energy, and (d) change in the average kinetic energy per atom, we can use the following formulas and assumptions:
(a) The work done by the gas can be found using the adiabatic work formula:
W = (γ / (γ - 1)) * P2V2 - P1V1
Where γ is the heat capacity ratio (for a monoatomic ideal gas, γ is 5/3), P1 and V1 are the initial pressure and volume, and P2 and V2 are the final pressure and volume.
(b) For an adiabatic process, there is no heat transfer, so Q = 0.
(c) The change in internal energy (ΔEint) can be found using the first law of thermodynamics, which states that ΔEint = Q - W.
(d) The change in average kinetic energy per atom (ΔK) can be related to the change in internal energy using the equation:
ΔEint = (3/2) * ΔK * N
Where N is the number of atoms.
Now let's solve the problem with the given information:
(a) Since we are given the change in temperature (ΔT), we can assume an initial and final temperature for the gas. Let's assume T1 = 300 K and T2 = 300 K + ΔT = 300 K + 15.0 K = 315 K.
(b) Since it is an adiabatic process, Q = 0.
(c) ΔEint = Q - W = 0 - W = -W
(d) To find ΔK, we need to know the number of atoms (N). Unfortunately, the number of atoms is not provided in the given information. Therefore, we cannot calculate ΔK without additional information.
Therefore, we can only find the value of (a) the work done by the gas. To find the work, we need the initial and final pressures and volumes of the gas. If we have additional information, we can use the adiabatic work formula mentioned above to calculate the work done.