A system consists of 4.8 mol of an ideal monatomic gas at 375 K.
(a) How much heat must be added to the system to double its internal energy at constant pressure?
? kJ
(b) How much heat must be added to the system to double its internal energy at constant volume?
? kJ
To solve 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 flow (Q) into the system minus the work (W) done by the system:
ΔU = Q - W
Since the question asks for the heat required, we can assume that the work done is zero because the conditions are given at constant pressure (for part a) and constant volume (for part b).
(a) To double the internal energy at constant pressure:
First, we need to determine the initial internal energy (U_initial) of the system. We can do this by using the ideal gas law:
PV = nRT
Where:
P = pressure
V = volume
n = number of moles
R = ideal gas constant
T = temperature
Rearranging the equation, we have:
V_initial = (nRT) / P
Now, we calculate the initial internal energy:
U_initial = (3/2) nRT
Next, we double the internal energy:
U_final = 2 U_initial
We can rearrange the equation to solve for the heat (Q):
Q = U_final - U_initial
Substituting the values:
Q = (2 U_initial) - U_initial
= U_initial
Q = (3/2) nRT
Since you mentioned that the system consists of 4.8 mol of gas and the temperature is given as 375 K, we can substitute these values into the equation:
Q = (3/2) (4.8 mol) (8.314 J/mol·K) (375 K)
= 14,130 J
To convert J to kJ, divide the value by 1000:
Q = 14,130 J / 1000
= 14.13 kJ
Therefore, the heat required to double the internal energy at constant pressure is 14.13 kJ.
(b) To double the internal energy at constant volume:
Since the conditions are given at constant volume, the work done (W) by the system is zero.
Therefore, the equation simplifies to:
ΔU = Q
We can use the same equation as in part (a) to calculate the initial internal energy (U_initial).
U_initial = (3/2) nRT
Then we can calculate the final internal energy (U_final):
U_final = 2 U_initial
Finally, we can calculate the heat required (Q):
Q = U_final - U_initial
Substituting the values:
Q = 2 U_initial - U_initial
= U_initial
Q = (3/2) nRT
Substituting the given values:
Q = (3/2) (4.8 mol) (8.314 J/mol·K) (375 K)
= 14,130 J
Converting J to kJ:
Q = 14,130 J / 1000
= 14.13 kJ
Therefore, the heat required to double the internal energy at constant volume is also 14.13 kJ.