# What exactly is the difference between temperature and heat? How is specific heat capacity related?

When I’ve searched online for the difference between temperature and heat, I’ve seen it defined as: Heat is the total kinetic energy of an object’s particles, whereas temperature is the average kinetic energy of an object’s particles. But that doesn’t make sense to me, because of two objections I have:

1. Heat is a form of energy and measured in joules. If temperature is the average kinetic energy per particle, then why is it an SI base unit and measured in kelvin and not some derived unit like joules/particle?
2. Specific heat capacity is the amount of heat energy it takes to raise a certain mass of a material by one kelvin. But if temperature is just the average kinetic energy of the molecules, why would different materials have different specific heat capacities? If a substance had x amount of joules per particle, wouldn’t it always take 1 joule per particle of substance to raise the average from x to x+1?

According to zeroth law

If two systems A and B, are separately in equilibrium with a third system, C then they are also in equilibrium with one another.

This gives empirical functions for A and B depending on the states of A and B such that

$$\Theta_1 (X_i)= \Theta_2 (Y_i)$$

If they are in thermal equilibrium the corresponding function becomes temperature and If they are in mechanical equilibrium this becomes force.

For an adiabatically insulated system the first law of thermodynamics says

The amount of work required to change the state of otherwise adiabatically insulated system depends only on the initial and final states, and not on the means by which the work is performed or intermediate stages.

So we can define a function let's say $$E(x_i,F_i)$$ where $$x_i$$ are generalized displacement and $$F_i$$ generalized forces which gives

$$dw=E(x_{i}^1,F_{i}^1)-E(x_{i}^2,F_{i}^2)$$

And let's call it internal energy and if suppose the system is not insulated then

$$dw=dE-dQ$$

(Kindly check the convention) In general we can write $$dw=\sum_i F_i dx_i$$

You see how temperature was analogue of a force in zeroth law,

Well it turns out using the 2nd law, we can arrive at $$dQ=TdS$$ (look it up and not giving you details) Which makes $$dE=Tds+\sum_i F_i dx_i$$

You can call temperature $$T$$ yet another generalized force with generalized displacement as entropy $$S$$.

So heat is related to temperature, as mechanical work is related to mechanical force.

Heat is a form of energy and measured in joules. If temperature is the average kinetic energy per particle, then why is it an SI base unit and measured in kelvin and not some derived unit like joules/particle?

The confusion you have is due to these "statement" being based on kinetic gas theory where one gets $$\langle E \rangle = \frac{3}{2} k_B T$$

This is only true for ideal monoatomic gas and the models based on it. In general it depends on the degree of freedom the particles have.

Like in case of solid one can assume the phonos making an ideal Bose gas and attribute this behaviour to phonons which are because of lattice vibration.

In thermodynamics heat is a form of energy that is transferred under a gradient of temperature, much like mechanical work is a for of energy that is transferred under a gradient of pressure. Temperature is a measure of the kinetic energy of molecules: the higher the temperature, the higher the kinetic energy. When molecules from a hot body collide with molecules from a colder one they transfer some of their energy to the colder system. Heat is the form of energy transferred via these collisions.

It is not correct to say

heat is the total kinetic energy of an object’s particles

Heat, like work, refers to an amount of energy transfer between two systems. Once transferred, this energy is stored as internal energy, which is the combination of kinetic and potential energy due to intermolecular interactions. Also, molecules with complicated structure can have several forms of "kinetic" energy, for example, rotational, vibrational and others, in addition to translational kinetic energy of the center of mass.

With respect to heat capacity, its definition is $$C_V = \left(\frac{\partial U}{\partial T}\right)_{V,N}$$ and tells us how internal energy is related to temperature under constant volume and constant number of particles. Writing this as $$dU = c_V dT \quad (\text{const.\ V, N})$$ it tells us that the amount of energy stored inside a fixed amount of matter at fixed volume increases with increasing temperature. In the special case that we are dealing with a monoatomic ideal gas, the only contribution to its energy comes from the translational kinetic energy and in this case (and this only) we obtain $$C_V=\frac{3}{2}k T$$.

Temperature is a thermodynamic property of a system; two properties, e.g. temperature and pressure, specify the state of a system. Temperature is an indication of the average kinetic energy of the molecules in the system.

Heat is energy transferred across the boundary of a system solely due to a temperature difference between the system and its surroundings. (Work is energy transferred across the boundary of a system due to a difference between any property except temperature between the system and its surroundings.)

It makes no more sense to talk about the heat in a body than it does to talk about the work in a body, although in mechanical engineering heat is sometimes use incorrectly in this context. The energy within a body is correctly called internal energy. Heat, work, and energy (including internal energy) are related through the first law of thermodynamics. Internal energy, like temperature, pressure, entropy, enthalpy, etc., is a property of the system. Heat (or work) added to a system increases its internal energy; the change in temperature, if any, is determined by the details of the process between the initial and final thermodynamic states.

Suggest you look at a good thermodynamics textbook, such as one by Sonntag and Van Wylen, for more details.

Heat is the total kinetic energy of an object’s particles, whereas temperature is the average kinetic energy of an object’s particles.

The total kinetic energy (KE) of an object's particles is the kinetic energy component of the internal energy of the object. It is not heat. Heat is the transfer of energy between objects due solely to a temperature difference between the object. Heat is energy in transit.

The consequences of energy transfer by heat can be an increase or decrease in the total molecular KE of the objects between which the the energy transfer occurs. Kinetic temperature is a measure of the average translational KE of the object's particles. See below.

1. Heat is a form of energy and measured in joules.

Again, heat is not a form of energy. It is a mechanism of transferring energy. The amount of energy transferred is measured in joules.

1. Specific heat capacity is the amount of heat energy it takes to raise a certain mass of a material by one kelvin. But if temperature is just the average kinetic energy of the molecules, why would different materials have different specific heat capacities?

Because the total KE of the particles is not necessarily just the translational KE associated with kinetic temperature. The total KE is potentially the sum of the translational, rotational, and vibrational KE's of the particles, depending on the type of particle.

For example, the specific heat of a diatomic gas is greater than a monatomic gas because in the case of the diatomic gas some of the heat absorbed can result in a change in vibrational and rotational KE's, in additional to translational KE, whereas a monatomic has only possesses translational KE. A change in the kinetic temperature of a gas is only a result of the change in average translational KE of gas molecules. Thus it take less heat to change the temperature of a monatomic gas, which has a lower specific heat, than a diatomic gas, which has a higher specific heat, all else being equal.