Question

If 8.40 kJ of heat is needed to raise the temperature of a sample of metal from 15 °C to 20 °C, how many kilojoules of heat will be required to raise the temperature of the same sample of metal from 25 °C to 40 °C?

Answer 1

`"25 kJ"`

Explanation:

The trick here is to realize that because the sample of metal has the same mass in both cases, you can say that

`q_2 = (DeltaT_2)/(DeltaT_1) * q_1`

Here

• `q_1` is the amount of heat needed to raise the temperature of the sample by `DeltaT_1 = 20^@"C" - 15^@"C"`
• `q_2` is the amount of heat needed to raise the temperature of the sample by `DeltaT_2 = 40^@"C" - 25^@"C"`

This equation can be found by using the fact that the heat absorbed by the metal can be calculated using the equation

`color(blue)(ul(color(black)(q = m * c * DeltaT)))`

Here

• `m` is the mass of the sample
• `c` is the specific heat of the metal

In your case, you can say that

`q_1 = m * c * DeltaT_1`

and

`q_2 = m * c * DeltaT_2`

Divide these two equations

`q_1/q_2 = (color(red)(cancel(color(black)(m * c))) * DeltaT_1)/(color(red)(cancel(color(black)(m * c))) * DeltaT_2)`

to get

`q_2 = (DeltaT_2)/(DeltaT_1) * q_1`

This equation tells you that in order to increase the temperature of the metal by a factor `(DeltaT_2)/(DeltaT_1)` when the mass of the metal is constant, the amount of heat supplied must also increase by a factor of `(DeltaT_2)/(DeltaT_1)`.

So, plug in your values to find

`q_2 = ((40 - 25) color(red)(cancel(color(black)(""^@"C"))))/((20-15)color(red)(cancel(color(black)(""^@"C")))) * "8.40 kJ"`

`color(darkgreen)(ul(color(black)(q_2 = "25 kJ")))`

The answer is rounded to two sig figs.