5  Thermodynamics & Fossil Fuels

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5.1 Fire!

5.2 Thermo Speak

5.4 Heat Capacities

5.5 Calorimetry

# 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.