A quantity of water at 20oC is mixed with another quantity of water at 70oC. The final steady temperature of…

Analyze the question: Cold water (20°C) is mixed with hot water (70°C), and the final temperature is 40°C. We need the ratio of the mass of cold water to the mass of hot water.

Concept: Assuming no heat is lost to the surroundings, the principle of heat exchange applies:

Heat gained by cold water = Heat lost by hot water

Heat = m c Δθ, and since both samples are water, the specific heat capacity c is the same and cancels out.

Solution:

Let m₁ = mass of cold water and m₂ = mass of hot water.

• Temperature rise of cold water = 40 − 20 = 20°C
• Temperature fall of hot water = 70 − 40 = 30°C

Heat gained = heat lost:

m₁ × c × 20 = m₂ × c × 30

m₁ × 20 = m₂ × 30

m₁/m₂ = 30/20 = 3/2

So the ratio of cold to hot is 3 : 2.

Tip: The ratio of masses is the inverse of the ratio of temperature changes. The cold water changes by 20 and the hot by 30, so the cold water mass is the larger one (30 parts) and the hot water mass is the smaller one (20 parts). This makes sense because the final temperature (40°C) is closer to the cold water's starting temperature, so there must be more cold water pulling the mixture toward it.