Two liquids of specific heat ratio 1:2 are at temperature `2theta and theta`

Options

A. if equal amounts of them are mixed, then temperature of mixture is `1.5 theta`
B. if equal amounts of them are mixed, then temperature of mixture is `4/3 theta`
C. for their equal amounts, the ratio of heat capacities is 1:1
D. for their equal amounts, the ratio of their heat capacities is 1:2.

Explanation

To solve the problem, we need to find the temperature of the mixture when two liquids with specific heat ratios of 1:2 are mixed, and then determine the ratio of their heat capacities.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Liquid 1 is at temperature \(2\theta\).
- Liquid 2 is at temperature \(\theta\).
- The specific heat ratio \( \frac{S_1}{S_2} = \frac{1}{2} \).

2. **Assume Mass and Specific Heat:**
- Let the mass of each liquid be \(M\).
- Let the specific heat of liquid 1 be \(S_1\) and that of liquid 2 be \(S_2\).
- From the ratio, we have \(S_1 = k\) and \(S_2 = 2k\) for some constant \(k\).

3. **Set Up the Heat Transfer Equation:**
- When the two liquids are mixed, the heat lost by liquid 1 must equal the heat gained by liquid 2.
- Heat lost by liquid 1:
\[ Q_1 = M \cdot S_1 \cdot (2\theta - T) \]
- Heat gained by liquid 2:
\[ Q_2 = M \cdot S_2 \cdot (T - \theta) \]

4. **Equate Heat Loss and Gain:**
\[ M \cdot S_1 \cdot (2\theta - T) = M \cdot S_2 \cdot (T - \theta) \]
- The mass \(M\) cancels out from both sides.

5. **Substitute the Specific Heats:**
\[ S_1 \cdot (2\theta - T) = S_2 \cdot (T - \theta) \]
- Substitute \(S_1 = k\) and \(S_2 = 2k\):
\[ k(2\theta - T) = 2k(T - \theta) \]

6. **Simplify the Equation:**
- Dividing both sides by \(k\) (assuming \(k \neq 0\)):
\[ 2\theta - T = 2(T - \theta) \]
- Expanding the right side:
\[ 2\theta - T = 2T - 2\theta \]

7. **Rearranging the Equation:**
- Move all terms involving \(T\) to one side:
\[ 2\theta + 2\theta = 2T + T \]
- This simplifies to:
\[ 4\theta = 3T \]

8. **Solve for \(T\):**
\[ T = \frac{4\theta}{3} \]

9. **Find the Ratio of Heat Capacities:**
- Heat capacity \(H_1\) of liquid 1:
\[ H_1 = M \cdot S_1 \]
- Heat capacity \(H_2\) of liquid 2:
\[ H_2 = M \cdot S_2 \]
- The ratio of heat capacities:
\[ \frac{H_1}{H_2} = \frac{S_1}{S_2} = \frac{1}{2} \]

### Final Answers:
- The temperature of the mixture \(T = \frac{4\theta}{3}\).
- The ratio of heat capacities \(H_1:H_2 = 1:2\).