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?

1 Answer
Jul 1, 2016

Answer:

#"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)|)))#