Definition
Amontons' law deals with the last combination of the three state variables. This refers to an isochoric state change. That means the volume remains constant, while temperature and pressure change.
Guillaume Amontons discovered that these two quantities are proportional to each other in the following way.
Law of Amontons
The ratio of the pressure and temperature of an ideal gas in an isochoric state change always remains constant.
\[
\frac{p}{T} = \text{const}
\]
Exercises
Exercise: Pressure in a Refridgerator
You open the refrigerator door for a relatively long time because you are looking for something. Then you close the door again. After some time, the appliance has returned to its normal temperature.
Now estimate what force you would need to open the door if the refrigerator were perfectly airtight by assuming the door as a perfect lever. The room temperature can be assumed as $20^\circ\,\mathrm{C}$, and the temperature inside the fridge as $5^\circ\,\mathrm{C}$. The dimensions of the fridge are $80\times 50\,\mathrm{cm}^2$
For the estimation, the following data are assumed:
The cooling of the air in a sealed refrigerator takes place at constant volume (isochoric process). This allows us to calculate the pressure inside the refrigerator after cooling with the help of Amonton's law:
$$
\frac{p_1}{T_1} = \frac{p_2}{T_2}
$$
$$
\Rightarrow p_2 = \frac{p_1 \cdot T_2}{T_1}
$$
$$
\Rightarrow p_2 = \frac{p_0 = 1013 ,\text{hPa} \cdot 278\,\text{K}}{293\,\text{K}} = 961\,\text{hPa}
$$
The pressure difference between inside and outside results in a force (F_{\rm P}) that acts at the center of the door:
$$
F_{\rm P} = \Delta p \cdot A
$$
$$
\Rightarrow F_{\rm P} = \left(1013 \cdot 10^2\,\text{Pa} - 961 \cdot 10^2\,\text{Pa}\right) \cdot 0.50\,\text{m} \cdot 0.80\,\text{m}
= 2100 ,\text{N}
$$
If the refrigerator door is treated as a single-sided lever, where the handle is twice as far from the hinge (axis of rotation) as the center of the door, then:
$$
F_{\rm Z} \approx \tfrac{1}{2} \cdot F_{\rm P} \Rightarrow F_{\rm Z} \approx 1000\,\text{N}
$$
This force corresponds approximately to the weight of a person with a mass of $100 ,\text{kg}$. Thus, if the refrigerator were completely airtight, it would be very difficult to open the door.