Three liquids A, B and C of specific heats `1cal//g-^@C, 0.5 cal//g-^@C and 0.25cal//g-^@C` are at temperatures `20^@C, 40^@C and 60^@C` respectively. Find temperature in equilibrium if they are mixed together. Their masses are equal.

Explanation

To find the equilibrium temperature when mixing three liquids A, B, and C with equal masses and different specific heats and temperatures, we can follow these steps:

### Step 1: Define the Variables
Let:
- Mass of each liquid = M (equal for all)
- Specific heat of liquid A, \( S_A = 1 \, \text{cal/g}^\circ C \)
- Specific heat of liquid B, \( S_B = 0.5 \, \text{cal/g}^\circ C \)
- Specific heat of liquid C, \( S_C = 0.25 \, \text{cal/g}^\circ C \)
- Initial temperatures: \( T_A = 20^\circ C \), \( T_B = 40^\circ C \), \( T_C = 60^\circ C \)
- Let \( T \) be the equilibrium temperature.

### Step 2: Write the Heat Transfer Equation
According to the principle of conservation of energy, the heat lost by the hotter liquids will be equal to the heat gained by the cooler liquids.

The heat lost or gained can be expressed as:
- For liquid A: \( Q_A = M S_A (T - T_A) = M \cdot 1 \cdot (T - 20) \)
- For liquid B: \( Q_B = M S_B (T - T_B) = M \cdot 0.5 \cdot (T - 40) \)
- For liquid C: \( Q_C = M S_C (T - T_C) = M \cdot 0.25 \cdot (T - 60) \)

### Step 3: Set Up the Equation
Since the total heat gained equals the total heat lost, we can set up the equation:
\[ Q_A + Q_B + Q_C = 0 \]
Substituting the expressions we derived:
\[ M (T - 20) + M (0.5)(T - 40) + M (0.25)(T - 60) = 0 \]
Dividing through by \( M \) (since \( M \) is non-zero):
\[ (T - 20) + 0.5(T - 40) + 0.25(T - 60) = 0 \]

### Step 4: Simplify the Equation
Expanding the equation:
\[ T - 20 + 0.5T - 20 + 0.25T - 15 = 0 \]
Combining like terms:
\[ (1 + 0.5 + 0.25)T - (20 + 20 + 15) = 0 \]
\[ 1.75T - 55 = 0 \]

### Step 5: Solve for T
Rearranging the equation gives:
\[ 1.75T = 55 \]
\[ T = \frac{55}{1.75} \approx 31.43^\circ C \]

### Final Answer
The equilibrium temperature when the three liquids are mixed is approximately \( 31.43^\circ C \).

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