Question

What is the specific heat capacity of a metal if it requires 178.1 J to change the temperature to 15.0 g of the metal from 25.00 C to 32.00 C?

Answer 1

`"1.70 J g"^(-1)""^@"C"^(-1)`

Explanation:

When a problem asks you to find a substance's specific heat, `c`, it's essentially telling you to find how much heat is needed in order to increase the temperature of `"1 g"` of said substance by `1^@"C"`.

In your case, you know that you need `"178.1 J"` in order to increase the temperature of `"15.0 g"` of your unknown metal by `7^@"C"`, since

`DeltaT = 32.00^@"C" - 25.00^@"C" = 7.00^@"C"`

So, how much heat would be needed to increase the temperature of `"1 g"` of this metal by `7.00^@"C ?"`

`1 color(red)(cancel(color(black)("g"))) * "178.1 J"/(15.0color(red)(cancel(color(black)("g")))) = "11.873 J"`

Since this much heat is needed to increase the temperature of `"1 g"` by `7.00^@"C"`, it follows that you can increase temperature by `1^@"C"` by adding

`1color(red)(cancel(color(black)(""^@"C"))) * "11.873 J"/(7.00color(red)(cancel(color(black)(""^@"C")))) = "1.70 J"`

Since you need `"1.70 J"` to increase the temperature of `"1 g"` of this metal by `1^@"C"`, you can say that the metal's specific heat will be

`c = color(green)(|bar(ul(color(white)(a/a)color(black)("1.70 J g"^(-1)""^@"C"^(-1))color(white)(a/a)|))) ->` rounded to three sig figs

ALTERNATIVELY

You can also use the following equation

`color(blue)(|bar(ul(color(white)(a/a)q = m * c * DeltaTcolor(white)(a/a)|)))" "`, where

`q` - the amount of heat released
`m` - the mass of the sample
`c` - the specific heat of the substance
`DeltaT` - the change in temperature, defined as the difference between the final temperature and the initial temperature

Rearrange to solve for `c`

`q = m * c * DeltaT implies c = q/(m * DeltaT)`

Plug in your values to find

`c = "178.1 J"/("15.0 g" * 7.00^@"C") = color(green)(|bar(ul(color(white)(a/a)color(black)("1.70 J g"^(-1)""^@"C"^(-1))color(white)(a/a)|)))`