Find out how much heat is needed to be supplied to a gas at a pressure of 1.25×105 Pa, such that the pressure increases by 25 per cent at constant volume, and the internal energy by 120 J.
To find out the amount of heat needed, we can use the first law of thermodynamics, which states that the change in internal energy (ΔU) of a closed system is equal to the heat (Q) supplied to the system minus the work (W) done by the system:
ΔU = Q - W
Since the volume (V) is constant in this scenario, the work done by the system is zero. Therefore, the equation simplifies to:
ΔU = Q
Given that the internal energy (ΔU) increases by 120 J, we know that the amount of heat (Q) supplied is also 120 J.
Now let's find the initial volume (V1) and initial pressure (P1) using the ideal gas equation:
P1V1 = nRT
Where:
- P1 is the initial pressure (1.25×105 Pa)
- V1 is the initial volume (unknown)
- n is the number of moles of gas (unknown)
- R is the ideal gas constant (8.314 J/mol·K)
- T is the temperature (unknown)
Since the number of moles (n) and temperature (T) remain constant, we can rewrite the equation as:
P1V1 = constant
Now, let's calculate the final pressure (P2) after a 25% increase:
P2 = P1 + (25/100)*P1 = P1 + 0.25*P1 = 1.25*P1
Using the constant relationship, we can also rewrite the ideal gas equation for the final state:
P2V2 = P1V1
Since P2 = 1.25*P1, the equation becomes:
1.25*P1*V2 = P1*V1
We can divide both sides by P1 to eliminate it:
1.25*V2 = V1
Now, since the volume is constant, V1 = V2 and we can rewrite the equation as:
1.25*V1 = V1
Therefore, V1 = V2.
Since V1 = V2 and ΔU = Q = 120 J, we can conclude that 120 J of heat is needed to be supplied to the gas at a pressure of 1.25×105 Pa in order to increase the pressure by 25% at constant volume and increase the internal energy by 120 J.