Laws Of Thermodynamics

Efficiency of Carnot engine: . If both and decrease by same value, ratio reduces → efficiency increases.

Step 1: Convert all quantities to consistent units (Joules)
We use the conversion factor: \SI{1}{cal} = \SI{4.184}{J} \begin{itemize} \item Initial internal energy ( ): \item Work done on the gas ( ): \item Heat released by the gas ( ): \end{itemize}
Step 2: Apply the First Law of Thermodynamics The First Law states: Substituting the values:
Step 3: Calculate the final internal energy
Step 4: Compare with the given options
The closest option to \SI{368.032}{J} is: \begin{center} Option 2: \SI{369.6}{J} \end{center} Final Answer: The final internal energy of the gas is \boxed{2}.

Step 1: Use the formula for heat added at constant pressure. where moles, , and .
Step 2: Use the relation between and .
Step 3: Calculate the increase in internal energy. The increase in internal energy is:

In an adiabatic process, there is no heat exchange with the surroundings, and the relation between the pressure, volume, and temperature of a gas is governed by the following equation: Where: - is the absolute temperature, - is the volume, - is the adiabatic index (ratio of specific heats ). This equation states that in an adiabatic process, the product of the temperature and volume raised to the power of is constant. Thus, the correct relation between volume and temperature for an adiabatic process is:

We will apply the principle of conservation of energy: the heat lost by the warm water will be equal to the heat gained by the ice. The energy required to bring the system to thermal equilibrium is split between:
The heat required to raise the temperature of the ice to 0°C.
The heat required to melt the ice.
The heat required to raise the temperature of the melted ice (now water) to the final equilibrium temperature.
Step 1: Calculate the heat required to raise the temperature of the ice from -10°C to 0°C.
The formula for heat is: Where:
is the mass of the ice,
is the specific heat of the ice,
is the temperature change.
For the ice:
Step 2: Calculate the heat required to melt the ice at 0°C.
The heat required to melt the ice is given by: Where:
is the latent heat of fusion of ice.
For the ice:
Step 3: Calculate the heat required to raise the temperature of the melted ice to the final equilibrium temperature.
Let the final temperature be (in °C). The heat required to raise the temperature of the melted ice from 0°C to is: Where: is the specific heat of water. For the melted ice:
Step 4: Calculate the heat lost by the water.
The heat lost by the warm water is: For the water:
Step 5: Apply conservation of energy.
The total heat gained by the ice is equal to the heat lost by the warm water: Substitute the values:
Step 6: Solve for .
Simplify the equation: Thus, the final temperature of the mixture is .

From thermodynamics: For adiabatic process,
Substitute in: