Question

A 77 g sample of water with a temperature of 30°C is added to 370 g water at 92°C in an insulated container. What is the final temperature after thermal equilibrium is reached?

Answer 1

The final temperature is 81.3 °C.

Explanation:

There are two heat transfers involved in this problem.

`"Heat gained by cold water + heat lost by hot water" = 0`

`q_1 + q_2 = 0`

The formula for the heat gained or lost by a substance is

`color(blue)(bar(ul(|color(white)(a/a)q = mcΔTcolor(white)(a/a)|)))" "`

∴ `m_1color(red)(cancel(color(black)(c)))ΔT_1 + m_2color(red)(cancel(color(black)(c)))ΔT_2 = 0`

Since `c` is a constant, we can cancel it and the equation becomes

`m_1ΔT_1 = m_2ΔT_2`

In this problem,

`m_1 = "77 g"`
`ΔT_1 = T_"f" - T_1^° = T_"f"color(white)(l) "- 30 °C"`
`m_2 = "370 g"`
`ΔT_2 = T_"f" - T_2^° = T_"f"color(white)(l)_1color(red)(cancel(color(black)(c)))ΔT_1 + m_2color(red)(cancel(color(black)(c)))ΔT_2 = 0`

`77 color(red)(cancel(color(black)("g"))) × (T_"f"color(white)(l) "- 30 °C) + 370"color(red)(cancel(color(black)("g")))× (T_"f"color(white)(l)"- 92 °C") = 0`

`77T_"f"color(white)(l)color(white)(l)"- 2310 °C" + 370T_"f"color(white)(l) "- 34 040 °C" = 0`

`447 T_"f" = "(2310 + 34 040) °C" = "36 350 °C"`

`T_"f" = "36 350 °C"/447 = "81.3 °C"`

The final temperature is 81.3 °C.