One gram of water is placed in a cylinder and the pressure is maintained at 2.0 x 105 Pa. The temperature of the water is raised by 31oC. Determine the change in internal energy when, a) the water is in its liquid phase and causes an expansion of 1.0 x 10-8 m3. b) Water is in its gaseous phase and expands by the much greater amount of 7.1 x 10-5 m3. Assume that there is no phase change for both cases. (cliquidH20 =4186 J/(kgoC); cgasH20 = 2020 J/(kgoC))
This looks like a situation where you should use the formula
dU = Qin - P dV
The heat in is Qin = C*dT. the two cases have different values of C and different values of the volume expansion, dV.
dU is the change in internal energy.
To determine the change in internal energy for both cases, we will use the equation:
ΔU = m * c * ΔT,
where:
ΔU is the change in internal energy,
m is the mass of the substance,
c is the specific heat capacity of the substance, and
ΔT is the change in temperature.
For part a), the water is in its liquid phase and causes an expansion of 1.0 x 10^-8 m^3. We are given the mass of the water, which is 1 gram. To convert this to kilograms, we divide by 1000:
m = 1 g / 1000 = 0.001 kg.
The specific heat capacity of liquid water (cliquidH20) is given as 4186 J/(kgoC), and the change in temperature (ΔT) is 31oC. Substituting these values into the equation:
ΔU = 0.001 kg * 4186 J/(kgoC) * 31oC.
Evaluating this expression, we find:
ΔU = 0.001 kg * 4186 J/(kgoC) * 31oC = 129.646 J.
Therefore, the change in internal energy for part a) is 129.646 J.
For part b), the water is in its gaseous phase and expands by 7.1 x 10^-5 m^3. The mass of the water and the specific heat capacity of water vapor (cgasH20) are not given, but we can assume they are the same as in part a) since there is no phase change.
Using the same mass as in part a) (0.001 kg), the specific heat capacity (2020 J/(kgoC)), and the change in temperature (31oC), we can calculate the change in internal energy using the same equation:
ΔU = 0.001 kg * 2020 J/(kgoC) * 31oC.
Evaluating this expression, we find:
ΔU = 0.001 kg * 2020 J/(kgoC) * 31oC = 62.62 J.
Therefore, the change in internal energy for part b) is 62.62 J.