Heat Capacities

"plain" Heat Capacity - C

The "plain" heat capacity is the quantity of heat that will cause a sample of substance (or even an entire object of some sort) to increase by one degree in temperature. Heat capacity is given the symbol \(C\) and has units of energy/temperature change which is J/°C or kJ/°C. Please remember that the "plain" heat capacity is referring to the "whole thing" - meaning the entire object and all its parts. You have to define your object and then figure out the heat capacity for the whole thing. The formula for heat (\(q\)) using "plain" heat capacity is

\[q = C \cdot \Delta T \]

If your sample or object never changes, the plain or whole heat capacity is the most useful choice of heat capacity.

Specific Heat Capacity - Cs

In real life, we don't always get the same amount of sample each time. The specific heat capacity allows us to vary the amount and account for with a mass term. For example, I might want 10 g of water, or 150 g... or 5000 g. My choice. So more often than not, we prefer to use a specific heat capacity for specific substances that we vary the amounts of - like water. Specific heat capacity is for a very specific amount (one gram) and the units are now J/g °C. This allows us to put mass (\(m\)) into our formula and have a much more versatile formula.

\[q = m \cdot C_{\rm s}\cdot \Delta T \]

Where \(m\) is mass in grams, \(C_{\rm s}\) is the specific heat capacity with units of J/g °C, and and \(\Delta T\) is the change in temperature in °C. It is specific heat capacity that we tend to list in tables of data. Different substances have different specific heat capacities. For example, water has a specific heat capacity of 4.184 J/g °C which, in the grand scheme of all things is a really high specific heat capacity - most other substances have lower values, like iron which has \(C_{\rm s}\) = 0.450 J/g °C. Many general chemistry students call this the "m-CAT" formula. If that rocks your boat, then use it.

Molar Heat Capacity - Cm

I would be remiss in my discussion here if I didn't also include a molar heat capacity - after all, we chemists tend to do so many things in moles of stuff - so why not have a heat capacity based on moles? We do and it is the molar heat capacity, \(C_{\rm m}\). The units are now J/mol °C. No surprise on the formula either...

\[q = n \cdot C_{\rm m} \cdot \Delta T \]

Where we now are using moles of substance (\(n\)) instead of mass.

An Example Heat Calculation

How many kilojoules (kJ) of heat must be applied to a cup of water at room temperature so that it reaches 40 °C ?

First off - "room temperature" is 25 °C. This means that we are increasing the temperature by 15 °C which is \(\Delta T\). One "cup" is 8 fluid ounces. We have to convert that to grams of water. After looking up a couple of conversion factors we have

(8 fl oz)(29.57 mL/oz)(1.00 g/mL) = 236.56 g of water

Now we can use our specific heat capacity formula...

q = m Cs ΔT
q = (236.56 g)(4.184 J/g°C})(15°C)
q = 14847 J = 14.8 kJ

Where are values for heat capacities? Well, they are all over the web - including lots and lots on Wikipedia. I have compiled a "short list" of many substances in Appendix 10.08 though to make things a bit easier. It is always best that we all use the same source for data on homeowork problems and the exams.

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