A metal jug of negligible thermal capacity contains a mass M of water at a 
temperature T1. A dry block of ice (specific latent heat of fusion = Lf) of mass m 
at 0 °C is dropped into the water (specific heat capacity = c) and begins to melt. 
At thermal equilibrium, there is no ice in the jug and the temperature of all the 
water is T2.
There is no thermal energy transfer between the atmosphere and the water or 
between the atmosphere and the ice.
Which expression is used to calculate m?

To calculate the mass \( m \) of the ice using the given information, we can apply the principle of conservation of energy. Since there is no heat exchange with the environment, the heat lost by the water must be equal to the heat gained by the ice.

### Step-by-step explanation:

1. **Heat lost by the water:**
   The water cools down from its initial temperature \( T_1 \) to the final equilibrium temperature \( T_2 \). The heat lost by the water is given by:

   \[
   Q_{\text{lost}} = M c (T_1 - T_2)
   \]
   where:
   - \( M \) is the mass of the water,
   - \( c \) is the specific heat capacity of water,
   - \( T_1 \) is the initial temperature of the water,
   - \( T_2 \) is the final equilibrium temperature.

2. **Heat gained by the ice:**
   The ice at 0°C melts and then the resulting water is warmed up to the equilibrium temperature \( T_2 \). The heat gained by the ice involves two processes:
   
   a. Melting of the ice:
   The heat required to melt the ice is:
   \[
   Q_{\text{melt}} = m L_f
   \]
   where \( L_f \) is the specific latent heat of fusion of ice, and \( m \) is the mass of the ice.
   
   b. Warming the melted ice water (now at 0°C) to the equilibrium temperature \( T_2 \):
   \[
   Q_{\text{warm}} = m c (T_2 - 0) = m c T_2
   \]

3. **Applying conservation of energy:**
   At equilibrium, the heat lost by the water is equal to the total heat gained by the ice:
   \[
   M c (T_1 - T_2) = m L_f + m c T_2
   \]

4. **Solving for \( m \):**
   Rearranging the equation to solve for \( m \):
   \[
   M c (T_1 - T_2) = m (L_f + c T_2)
   \]

   Finally, the expression for the mass \( m \) of the ice is:
   \[
   m = \frac{M c (T_1 - T_2)}{L_f + c T_2}
   \]

### Summary:
The expression to calculate the mass \( m \) of the ice is:
\[
m = \frac{M c (T_1 - T_2)}{L_f + c T_2}
\]

Let me know if you need further clarification!

### Related Questions:
1. What is the specific latent heat of fusion, and how does it affect melting?
2. How does the specific heat capacity of water influence temperature changes?
3. Can this method be used if the ice doesn’t fully melt? Why or why not?
4. What would happen if there was heat exchange with the surroundings?
5. How does the final equilibrium temperature \( T_2 \) change if the initial temperature \( T_1 \) increases?

### Tip:
Always check the units of all quantities (mass, temperature, latent heat, and heat capacity) to ensure they are consistent when solving thermal energy problems.

A metal jug of negligible thermal capacity contains a mass M of water at a temperature T1. A dry block of ice (specific latent heat of fusion = Lf) of mass m at 0 °C is dropped into the water (specific heat capacity = c) and begins to melt. At thermal equilibrium, there is no ice in the jug and the temperature of all the water is T2. There is no thermal energy transfer between the atmosphere and the water or between the atmosphere and the ice. Which expression is used to calculate m?

Learn how to calculate the mass of ice dropped into water using the principle of energy conservation. This problem covers key thermodynamic concepts, including specific heat capacity and latent heat of fusion. Suitable for students in Grades 10-12.

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