Units in a heat transferred calculation

[smulc]

I'm trying to do a calculation using: Heat transferred = mass x specific heat capacity x change in temperature
q=mc∆T

But the value I'm trying to calculate is the change in temperature so I've rearranged the equation to ∆T= q/mc

I think the numbers and answer I have are correct, but the units are confusing me. My answer clearly needs to be in °C, the joules on the top and the bottom will cancel out but then on the bottom I'm left with kg multiplied by kg^-1.

3.54 J
450 kg x 4.2 x 10³ J kg^-1 °C^-1

As far as I know, these don't cancel each other out so my answer has the wrong units. I don't know if I've done the entire thing wrong if I'm just confusing myself over something silly. I'd appreciate any help at all.

 

[Redbelly98]

... on the bottom I'm left with kg multiplied by kg^-1.

So are you wondering what to do with kg·kg^-1?

kg is the same as kg⁺¹, so what you really have is kg⁺¹·kg^-1. Simplify that using algebra rules for exponents, and you should be all set.

 

[smulc]

ohh I feel really stupid now. Any number to the zero power is equal to 1. I already knew this but mistakingly thought that the result of calculating kg⁰ would be 1kg, but the answer is literally just 1, isn't it? So the kg does actually cancel. I feel silly for not realising, thanks very much for the help!

 

[Redbelly98]

You're welcome, glad it worked out for you.

 

[asteriatic]

Hello,

Thank you for sharing your calculations and concerns. It seems like you have a good understanding of the equation for heat transferred (q=mc∆T) and how to rearrange it to solve for the change in temperature (∆T= q/mc). However, you are correct that the units for your final answer do not match the desired units of °C.

To solve this issue, you will need to pay attention to the units as you rearrange the equation. Let's break down the units for each term in the equation:

- q (heat transferred) has units of joules (J)
- m (mass) has units of kilograms (kg)
- c (specific heat capacity) has units of joules per kilogram per degree Celsius (J/kg°C)
- ∆T (change in temperature) has units of degrees Celsius (°C)

When you rearrange the equation to solve for ∆T, you will need to make sure that the units on the top and bottom of the equation cancel out, leaving you with only the desired units for ∆T. Let's see how this works:

∆T = q/mc

- q has units of J and is on the top of the equation. On the bottom, we have m (kg) multiplied by c (J/kg°C). This means that the units on the bottom will cancel out, leaving us with only J on the top.
- Since we want our final answer to be in °C, we need to make sure that the units on the top are also in °C. To do this, we can use the specific heat capacity (c) to convert the units of J to °C. Remember, c has units of J/kg°C, which means that it tells us how many joules are needed to raise the temperature of 1 kg of a substance by 1 °C. So, we can divide our answer in J by c to get the units in °C.
- This gives us the final equation: ∆T = q/c. The units for q (J) cancel out with the units for c (J/kg°C), leaving us with only the desired units of °C for ∆T.

Applying this to your specific example:

∆T = 3.54 J / (450 kg x 4.2 x 10^3 J/kg°C)

The units for J cancel out, leaving us with only