Kinetic Theory

Kinetic theory states that at a microscopic level all matter is made from particles. They are always in motion and attracted to each other through the IMF. Because they are in motion they have kinetic energy, and because they have attractive forces they have potential energy.

Adding or removing energy can either change the distance between the particles (PE) or the speed of the particles (KE). The internal energy is the sum of the kinetic energy and potential energy

$u = KE + PE$

Kelvin: The SI unit of temperature. $0 C = 273.15 K$, $\Delta C = \Delta k$

Density is a substance’s mass per unit volume. $P = \frac{m}{v}$ in either $kg m^{-3}$ or $g cm ^{-3}$

Temperature Time Graphs

The moment where the graph is rising is an increase of kinetic energy with the temperature of the object. Given this is a closed container this will increase linearly. When the graph is in a straight line is when the potential energy is increasing in the form of a phase shift.

A graph of the temperature over a function of time is called a heating curve and will be around the same for every substance

Specific Heat Capacity

The specific heat capacity of a substance is the amount of energy required to heat 1 $kg$ of a substance by 1 degree celsius or kelvin (remember change in celsius is the same as change in kelvin).

$c = \frac{Q}{m \Delta T}$

Units of $J kg^{-1} K^{-1}$

Q is thermal energy. M is the mass of the object. C is the specific heat capacity. Delta T is the change in temperature (k or c).

The phase a substance is in changes its specific heat capacity

Specific Latent Heat

The specific latent heats of fusion and vaporization are the energy needed to change the phase of 1 kg of a substance at its melting (fusion) or boiling point (vaporization).

$L = \frac{Q}{m}$

$L_f$ or $L_v$

Units of $J kg^{-1}$

Q is thermal energy. M is mass. $L_f$ is specific latent heat of fusion while $L_v$ is specific latent heat of vaporization

Note that fusion is the same as melting, and vaporization is the same as liquidating from a gaseous state

The gradients/slopes at the graph where kinetic energy is increasing is quantified by the formula $Q=m c \Delta t$. The gradient of the line is the change in temperature over time, and objects with a higher specific heat capacity will need more energy to heat up, in turn decreasing the gradient.

When the object is going through a phase change the temperature of the line is staying the same, and the gradient is 0. $Q = mL$ applies here. The form with the highest specific latent heat will have a longer line. Keep in mind the specific latent heat of vaporization is always higher than the specific latent heat of fusion.

Two objects that are touching each other will transfer energy until they are in thermal equilibrium (at the same temperature). Due to the conservation of energy, the thermal energy lost by the otter one will be the same as the thermal energy gained by the colder one.

For questions that look like that, use the formula

$Q_{out} = Q_{in}$

$-m_1 c_w \Delta T = m_2 c_w \Delta T$

It is negative because the one losing temperature has a negative thermal energy

Gas Laws

Gasses can be described by their pressure, temperature, and volume. The temperature of a gas is directly proportional to its average kinetic energy. Pressure of a gas is the force of a gas against a container's walls. The volume of a gas is the space occupied by a gas. Because of the collisions between particles, this is the volume of the container it is in.

Ideal Gasses

Particles are identical

Particles spacing are much larger than the particles themselves

No intermolecular forces

Elastic collision

In other words, if they have a high temperature, low pressure, and low density they can be considered ideal in this scope

Moles

1 mole is $6.02 * 10^{23}$ particles

The molar mass is the mass of one mole of a substance

Pressure

Pressure = $F/A$

Pascal = $N m^{-2}$

Pressure = $\frac{1}{3} \rho \overline{v}^2$

Because the IMF of ideal gasses is negligible, all ideal gasses have a PE of 0. This means that the internal energy is just the kinetic energy.

$\overline{E}_k = \frac{3}{2} k_b T$

Kinetic energy average = 3/2 Boltzmann's constant times temperature

Therefor, the internal energy (total kinetic energy) can be determined by the average kinetic energy multiplied by the number of particles

$u = \frac{3}{2} k_b T N$

Internal Energy = 3/2 Boltzmann's constant times temperature times number of particles

You can also represent this using a different formula using the number of moles

$u = \frac{3}{2} R T n$

Internal Energy = 3/2 gas constant times temperature times number of moles

Keep in mind that

$K_b = \frac{R}{N_a}$

Gas Laws

First Law of Thermodynamics

$Q = \Delta U + w$

Q is thermal energy. Delta U is the change in internal energy. W is the work done by the gas.

If thermal energy enters a system, Q is set as a positive value, while leaving makes Q a negative value.

The increase in temperature leads to a positive delta U while a decrease in temperature leads to a negative delta U

If the volume expands the work is positive, and if it decreases the work is negative. It is represented by the volume under a pressure volume gas.

Pressure Volume Graphs