# $Q=mc\Delta t$ vs $Q=c\Delta t$

The thermal capacity of a $60 \;\mathrm{kg}$ human is $210 \;\mathrm{kJ/°C}$. How much heat is lost from a body if its temperature drops by $2\;\mathrm{ °C}$?

My original working out was: $$Q=mc\Delta{t}$$ $$Q=(60)(210000)(2)\;\mathrm{J}$$ $$Q=25200000 \;\mathrm{J}$$

However, from definition, $$Q = c\Delta{t}$$ $$Q = (210000)(2)\;\mathrm{J}$$ $$Q = 420000\;\mathrm{J}$$ And this second answer is the one given in the textbook. Why do we not take into account mass for such a question?

• Besides all the correct answers, I might point out that keeping your units in the calculation would help. Your first working out should not yield an answer in $J$. – BowlOfRed Oct 27 '15 at 0:28

## 3 Answers

Here you are confusing heat capacity $C$ and specific heat capacity $c = C/m$. The question gives you the heat capacity. You can tell because it is in $kJ/^o C$, not $kJ/(kg\;^oC)$.

This is why you should always include units in your calculations. In the first calculation, you would have gotten an answer with units of mass*energy instead of energy, and you would have seen your mistake right away.

• Looks like two people beat me to it. Whoops! – Matt Dickau Oct 26 '15 at 23:53
• I guess we all beat each other to it. Three answers within one minute... – Steeven Oct 26 '15 at 23:55
• How is the heat capacity useful if mass is a key factor that influences the heat input or removal required to change the temperature? Does this mean that both answers provided in my question are incorrect? – GoodChessPlayer Oct 27 '15 at 0:54
• No, the second calculation you did is the correct one. The influence of mass is included within the heat capacity - something with a higher mass $m$ made out of the same material has a higher heat capacity $C$ (because it has the same specific heat capacity $c$, so the product $C=mc$ is higher). – Matt Dickau Oct 27 '15 at 1:07

You have encountered the difference between specific heat capacity and heat capacity. Heat capacity refers to the heat input or removal required to change the temperature of a certain mass of material (in your case, 60 kg of humanity) by 1 temperature unit. 'Specific heat capacity' refers to the heat input or removal per unit mass of a material required to change the temperature by 1 unit. They are similar, but not the same.

In your case, the specific heat capacity would work out to be $\frac{210}{60}\,\frac{kJ}{kg\cdot\,^o C}= 3500\,\frac{kJ}{kg\cdot\,^o C}$. This agrees with the values given at several websites.

EDIT: In fact, when one does a heat/temperature experiment on an object, the ratio of the heat, $Q$, to the temperature change, $\Delta T$, is the heat capacity of that object. If the object is of uniform material (water, brass, nickel alloy, uniform plastic, etc), we assume (with good reasons) that every nanogram (or microgram, etc) will change temperature in an identical fashion as every other nanogram. With that assumption, we take that ratio and divide by the mass to get a material-based behavior, supposedly independent of mass. Numerous experiments have confirmed this behavior. On the other hand, if the object is not wholly of one material, dividing the heat capacity by the mass doesn't make much sense unless one is dealing with other object which have the same mixture of materials. For example, a 60 kg person with low fat content and high muscle content will have a different heat capacity than a 60 kg person with high fat content. The specific heat of muscle is generally higher than the specific heat of fat. See this [tissue database].1

• How is the heat capacity useful if mass is a key factor that influences the heat input or removal required to change the temperature? Does this mean that both answers provided in my question are incorrect? – GoodChessPlayer Oct 27 '15 at 0:54
• Heat capacity is useful if the object is a fixed conglomeration of different objects and/or materials. It can give you an idea of how other similar objects may behave. There is an edit in my answer, too. – Bill N Oct 27 '15 at 15:01

You are mixing up heat capacity $C$ with specific heat capacity $c$:

$$C=mc$$

$c$ is the heat capacity per mass (in joules per degree per kilogram, $\mathrm{[\frac{J}{^\circ C \cdot kg}]}$), while $C$ is the overall heat capacity for the object as a whole (in joules per degree, $\mathrm{[\frac{J}{^\circ C}]}$). The expressions should look like this:

$$Q=mc\Delta T=C\Delta T$$

In the question, you see from the units that you are given $C$, not $c$.

• How is the heat capacity useful if mass is a key factor that influences the heat input or removal required to change the temperature? Does this mean that both answers provided in my question are incorrect? – GoodChessPlayer Oct 27 '15 at 0:55

## protected by Qmechanic♦Oct 26 '15 at 23:52

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