A liquid of mass m and specific heat c is heated to a temperature 2T. Another liquid of mass m/2 and specific heat 2 c is heated to a temperature T. If these two liquids are mixed, the resulting temperature of the mixture is

Options

A. `(2//3)T`
B. `(8//5)T`
C. `(3//5)T`
D. `(3//2)T`

Explanation

To solve the problem of finding the resulting temperature when two liquids are mixed, we can use the principle of conservation of energy. The heat gained by one liquid will be equal to the heat lost by the other liquid.

### Step-by-Step Solution:

1. **Identify the parameters of the two liquids:**
- For the first liquid:
- Mass (m1) = m
- Specific heat (c1) = c
- Initial temperature (T1) = 2T
- For the second liquid:
- Mass (m2) = m/2
- Specific heat (c2) = 2c
- Initial temperature (T2) = T

2. **Use the formula for the final temperature (Tf) of the mixture:**
The formula for the final temperature when two substances are mixed is given by:
\[ T_f = \frac{m_1 c_1 T_1 + m_2 c_2 T_2}{m_1 c_1 + m_2 c_2} \]

3. **Substitute the values into the formula:**
- Calculate the numerator:
\[ m_1 c_1 T_1 = m \cdot c \cdot (2T) = 2m c T \]
\[ m_2 c_2 T_2 = \frac{m}{2} \cdot (2c) \cdot T = m c T \]
\[ \text{Total numerator} = 2mcT + mcT = 3mcT \]

- Calculate the denominator:
\[ m_1 c_1 = m \cdot c = mc \]
\[ m_2 c_2 = \frac{m}{2} \cdot (2c) = mc \]
\[ \text{Total denominator} = mc + mc = 2mc \]

4. **Calculate the final temperature:**
\[ T_f = \frac{3mcT}{2mc} = \frac{3}{2} T \]

5. **Conclusion:**
The resulting temperature of the mixture is:
\[ T_f = \frac{3}{2} T \]

### Final Answer:
The resulting temperature of the mixture is \( \frac{3}{2} T \).

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