Two equal amounts of water are mixed by-gently pouringboth into an insulated cup. One part is initially `90^@ C` , andthe other part is initially at `T_i^@ C`. If the final temperature of the mixture is `131^@ C`, what is the value of T

Options

A. `90^@`
B. 68
C. 20
D. 0

Answer

To solve the problem of mixing two equal amounts of water at different temperatures, we can use the principle of conservation of energy. The heat lost by the hotter water will be equal to the heat gained by the cooler water.

### Step-by-Step Solution:

1. **Identify the Variables:**
- Let the mass of each amount of water be \( m \).
- The initial temperature of the first part of water, \( T_1 = 90^\circ C \).
- The initial temperature of the second part of water, \( T_i \) (unknown).
- The final temperature of the mixture, \( T_f = 131^\circ C \).

2. **Write the Heat Transfer Equation:**
- According to the conservation of energy, the heat lost by the first part of water equals the heat gained by the second part:
\[ q_1 + q_2 = 0 \]
where:
- \( q_1 \) is the heat lost by the first part of water.
- \( q_2 \) is the heat gained by the second part of water.

3. **Express the Heat Transfer:**
- The heat lost by the first part of water can be expressed as:
\[ q_1 = m \cdot c \cdot (T_f - T_1) = m \cdot c \cdot (131 - 90) \]
- The heat gained by the second part of water can be expressed as:
\[ q_2 = m \cdot c \cdot (T_f - T_i) = m \cdot c \cdot (131 - T_i) \]
Here, \( c \) is the specific heat capacity of water, which cancels out since both masses are equal.

4. **Set Up the Equation:**
- Substituting the expressions for \( q_1 \) and \( q_2 \) into the conservation of energy equation:
\[ m \cdot c \cdot (131 - 90) + m \cdot c \cdot (131 - T_i) = 0 \]
- Simplifying this gives:
\[ (131 - 90) + (131 - T_i) = 0 \]

5. **Solve for \( T_i \):**
- Simplifying further:
\[ 41 + (131 - T_i) = 0 \]
- Rearranging gives:
\[ 131 - T_i = -41 \]
- Thus:
\[ T_i = 131 + 41 = 172^\circ C \]

6. **Conclusion:**
- Therefore, the initial temperature \( T_i \) of the second part of water is \( 172^\circ C \).

### Final Answer:
The value of \( T_i \) is \( 172^\circ C \).