A

To find the temperature of the mixture when equal volumes of three liquids are mixed, we can use the principle of conservation of energy due to heat transfer.

Step 1: Write the equation for heat loss and heat gain. Given that the liquids are in thermal equilibrium when they mix, the heat lost by the hotter liquids will be equal to the heat gained by the cooler liquids.

Step 2: The heat gained/lost by each liquid can be expressed as:

• For liquid 1: Q_1 = m_1 c_1 (T_f - t_1)
• For liquid 2: Q_2 = m_2 c_2 (T_f - t_2)
• For liquid 3: Q_3 = m_3 c_3 (T_f - t_3)

where T_f is the final temperature of the mixture, m_i (mass) = V * ρ_i (density), and V is the equal volume assumed for all.

Step 3: Since the volumes are equal, we can express the masses as:
m_1 = V * ρ_1, m_2 = V * ρ_2, m_3 = V * ρ_3
Step 4: Setting the sum of heats to zero gives us:
V * ρ_1 c_1 (T_f - t_1) + V * ρ_2 c_2 (T_f - t_2) + V * ρ_3 c_3 (T_f - t_3) = 0
Step 5: Simplify:
ρ_1 c_1 (T_f - t_1) + ρ_2 c_2 (T_f - t_2) + ρ_3 c_3 (T_f - t_3) = 0
Step 6: Expanding this equation results in:
ρ_1 c_1 T_f - ρ_1 c_1 t_1 + ρ_2 c_2 T_f - ρ_2 c_2 t_2 + ρ_3 c_3 T_f - ρ_3 c_3 t_3 = 0

Step 7: Collect terms involving T_f:
(ρ_1 c_1 + ρ_2 c_2 + ρ_3 c_3) T_f = ρ_1 c_1 t_1 + ρ_2 c_2 t_2 + ρ_3 c_3 t_3
Step 8: Therefore, the final temperature T_f can be derived as:
T_f = \frac{ρ_1 c_1 t_1 + ρ_2 c_2 t_2 + ρ_3 c_3 t_3}{ρ_1 c_1 + ρ_2 c_2 + ρ_3 c_3}

Thus, the final temperature of the mixture can be expressed in terms of the individual properties of the liquids and their respective initial temperatures.