To determine the work done on the thermally insulated system, we need to analyze the energy changes that occur when the vessel is shaken. The key here is to understand that the work done on the system results in an increase in the internal energy of the water, which in turn raises its temperature. Let's break this down step by step.
Understanding the Energy Transfer
In a thermally insulated system, the heat exchange with the surroundings is negligible. Therefore, any increase in temperature must come from the work done on the system. The formula we can use to calculate the change in internal energy due to the temperature rise is:
Calculating the Heat Gained by Water
The heat gained by the water can be calculated using the formula:
Where:
- Q = heat gained (in Joules)
- m = mass of the water (in kg)
- c = specific heat capacity of water (in J/(kg·K))
- ΔT = change in temperature (in K or °C)
Plugging in the Values
Given:
- Mass of water, m = 200 g = 0.2 kg
- Specific heat capacity of water, c = 4200 J/(kg·K)
- Initial temperature, T1 = 15 °C
- Final temperature, T2 = 17 °C
The change in temperature, ΔT, is:
- ΔT = T2 - T1 = 17 °C - 15 °C = 2 °C
Now, substituting these values into the heat equation:
- Q = (0.2 kg)(4200 J/(kg·K))(2 K)
- Q = 0.2 * 4200 * 2 = 1680 J
Calculating Work Done on the System
Since the system is thermally insulated, the work done on the system is equal to the heat gained by the water:
Thus, the work done on the system is:
Final Thoughts
In summary, the work done on the thermally insulated copper vessel containing water, which caused the temperature to rise from 15 °C to 17 °C, is 1680 Joules. This calculation illustrates how mechanical work can convert into thermal energy within a closed system, leading to an increase in temperature without any heat exchange with the environment.
