# Is $Q = m C \Delta T$ the equation of heat or thermal energy?

I'm little bit confused about that equation as I know that heat is the energy that is transferred through convection, conduction, and radiation one of which has its own equation whereas thermal energy (sensible or latent energy) is one of internal energy form. Give me some information related those please.

The equation

$$Q=mC\Delta T$$

tells how much energy $$Q$$ that is needed to raise the temperature with $$\Delta T$$ of the object (of mass $$m$$ and with heat capacity $$C$$) in question.

That energy may come in different forms. This equation does not say anything about the form of energy, only about the needed amount of energy.

Sure, the energy will be absorbed and converted into thermal energy in the object (because energy that raises a temperature is what we call thermal). But this does not tell anything about what form the energy was before. It could be thermal energy before (heat up the object in an oven), it could be solar energy (let radiation hit and heat up the object), it could be electrical (generate the energy within the object via electric/magnetic induction or the like, if the object is conductive) etc.

• Thank you so much for your answer. I still have a doubt though, how is the relation between equation Q = mc delta t and equation of heat transfer rate? conduction for example, the rate of heat by this process is explained by dQ/dt = −𝑘𝐴((𝑇_2−𝑇_1)/𝐿) Mar 3, 2020 at 9:37
• @Elluthfi While $$Q = mC \Delta T$$ tells how much energy that is needed in total (regardless of time), a heat transfer equation tells how fast energy is transferred. Use the above equation to calculate the total amount of needed energy, and then use $$\frac{dQ}{dt} = −\kappa 𝐴\frac{\Delta T}L\quad\Leftrightarrow\quad dQ = −\kappa 𝐴\frac{\Delta T}L\;dt \quad\Leftrightarrow\quad Q = \int_0^{t_\text{total}} −\kappa 𝐴\frac{\Delta T}L\;dt$$ to find the total time $t_\text{total}$ it takes to accumulate that amount of energy. Mar 3, 2020 at 9:59
• @elluthfi A side note: Note that you can make math formatting here on the SE site by wrapping your math in Dollar signs \$...\$ or \$\$...\$\$. Mar 3, 2020 at 10:00
• Ok thank you so much for your input Mar 3, 2020 at 11:57
• @elluthfi No problem. By the way, I can see that there is a small ambiguity in the comment I wrote. The $\Delta T$ in the $Q=mC\Delta T$ equation is the temperature difference that you raise the temperature with. The $\Delta T$ in the $\frac{dQ}{dt} = −\kappa 𝐴\frac{\Delta T}L$ equation is the difference in temperature between where the heat comes from (the source) and where it flows to (the object). So not the same temperature difference. Just to clarify. Mar 3, 2020 at 12:11

In thermodynamics, we start out by subdividing the world into two entities, the system under consideration and everything else, which we call the surroundings. There is a boundary interface between these two entities is where physical work is done and thermal energy (heat) passes. When such work and heat exchange (between the system and surroundings) take place, we call this a thermodynamic process. We call the cumulative amount of heat passing through the boundary during a process Q and the work W. According to the first law of thermodynamics, the difference between the heat added to the system and the work done by the system on its surrounding is equal to the change in internal energy of the system $$\Delta U$$, where U represents the total kinetic- and interactional potential energy of the molecules comprising the system. If the process takes place at constant volume so that no work is done (W=0), then the change in internal energy is just equal to the heat added: $$\Delta U=Q$$. Since the internal energy of the system depends on the temperature T and volume V of the system, we can represent the change in internal energy for a constant volume process by $$\Delta U=mC_v\Delta T$$, where m is the mass of the system, Cv is the heat capacity of the system (a physical property of the material), and $$\Delta T$$ is the change in temperature. Thus, for a constant volume process, we also have that $$Q=mC_v\Delta T$$.

For more general processes we write that $$dU=\left(\frac{\partial U}{\partial T}\right)_VdT+\left(\frac{\partial U}{\partial V}\right)_TdV=mC_vdT+\left(\frac{\partial U}{\partial V}\right)_TdV$$That is, we define the heat capacity of the material comprising the system in terms of the partial derivative of its internal energy with respect to temperature as: $$C_v=\frac{1}{m}\left(\frac{\partial U}{\partial T}\right)_V$$This relationship applies in general. However, for a process that does not take place at constant volume, so that work is done, $$Q\neq m C_v\Delta T$$.

In addition to all this, there is a terminology issue that causes confusion among students. This is because the symbol Q is sometimes used, not to represent the cumulative amount of heat transferred to a system across its boundary with its surroundings but also to represent the #rate* of heat transfer. In this case, for conductive heat transfer across the boundary, the rate of conductive heat transfer is modeled as $$-kA\frac{dT}{dx}$$, where k is the thermal conductivity of the material comprising the system, A is the heat transfer area, and dT/dx is the temperature gradient normal to the boundary interface. For convective heat transfer, it is modeled as $$hA(T_{surr}-T)$$ where h is the heat transfer coefficient and $$T_{surr}$$ is the temperature of the surroundings at the boundary interface.

• What a comprehensive explanation, thank you very much Mar 6, 2020 at 0:02

In the equation in your title $$\Delta T$$ is the change in temperature of a substance resulting from energy transferred to or from that substance which may be in the form of heat or work.

In heat transfer rate equations for conduction and convection $$\Delta T$$ Is the difference in temperature between substances or regions of a substance where heat transfer is occurring and that drives heat transfer.

Hope this helps

• I see, that's what i'm still confused too. Thank you Mar 3, 2020 at 12:11
• @elluthfi what are you still confused about? Mar 3, 2020 at 12:16
• The relation between those two equations (rate of conductive heat transfer and sensible heat) but, thanks to you all,i already got much insight Mar 5, 2020 at 23:58

I think that you are really asking about the difference between heat and thermal energy as in the equation you have quoted $$Q$$ can be the heat, energy transferred from one system to another colder system, or the change in thermal energy of a system. The amount of transfer or change can be gauged by measuring the temperature change, $$\Delta T$$.

Energy in transfer can include conduction, radiation, electrical heating, mechanical stirring, gravitational work (a famous story about Joule is that he used part of his honeymoon to measure the differences in water temperature at the top and bottom of a waterfall), etc.

One definition if thermal energy is that it is the internal energy of a system but people do use thermal energy and heat as meaning the same thing.

• Thank you so much although what i read in a textbook, such as what is written by michael a boles and yunus a cengel, they differ between heat, and thermal energy as the thermal energy is included in internal energy (sensible and latent energy) Mar 3, 2020 at 12:07