Ideal Gas Law: When Is It Applicable?

The ideal gas law is a good approximation of the behaviour of many gases under certain conditions. It is often written in the form: pV=nRT, where p is the absolute pressure of the gas, V is the volume of the gas, n is the amount of substance of gas (or number of moles), R is the ideal gas constant, and T is the absolute temperature of the gas.

The ideal gas law is applicable when the pressure is low and the temperature is high. It is a good approximation when the pressure is below 0.1 and the temperature is above 1.819 - (0.3546/pressure^0.6). It is not a good approximation when the pressure is above 7, regardless of the temperature.

The ideal gas law can be used to calculate pressure change, temperature change, volume change, or the number of molecules or moles in a given volume.

Characteristics	Values
When to use	The ideal gas law is used when the amount of gas is given and the mass of the gas is constant.
When not to use	The ideal gas law does not apply to liquids.
Best conditions	The ideal gas law is a good approximation when \(P_r<0.1\) (even for a saturated vapour). It is also a good approximation when \(0.1<P_r<7\) if \(T_r>1.819-\frac{0.3546}{P_{\!r}^{\,0.6}}\,\).
Worst conditions	The ideal gas law is not a good approximation when \(P_r>7\), no matter the temperature.
Pressure	The law applies perfectly when \(P\rightarrow 0\).
Temperature	The ideal gas law is best applied at high temperatures.
Assumptions	The ideal gas law assumes point particles with no volume, non-interacting particles, and random motion.

What You'll Learn

Calculating pressure changes due to temperature changes
The ideal gas law and energy
When the ideal gas law is applicable?
The ideal gas law and the combined gas law
The ideal gas law in thermodynamic processes

Calculating pressure changes due to temperature changes

The ideal gas law is a useful tool for understanding the relationship between the pressure, volume, and temperature of a gas. It is expressed as:

PV = NkT

Where P is the absolute pressure of a gas, V is the volume it occupies, N is the number of atoms and molecules in the gas, T is its absolute temperature, and k is the Boltzmann constant.

The ideal gas law can be used to calculate pressure changes due to temperature changes. For example, let's consider a bicycle tire that is fully inflated with an absolute pressure of 7.00 x 10^5 Pa at a temperature of 18.0°C. We want to calculate the pressure after the temperature has risen to 35.0°C, assuming no leaks or changes in volume.

First, we need to identify the known values:

Initial pressure (P0) = 7.00 x 10^5 Pa
Initial temperature (T0) = 18.0°C
Final temperature (Tf) = 35.0°C

We can then use the ideal gas law equation to solve for the final pressure (Pf):

P0V0 = NkT0

PfVf = NkTf

Since the volume and the number of atoms are constant, we can divide PfVf by P0V0 to get:

Pf/P0 = Tf/T0

Rearranging this equation, we get:

Pf = P0 * Tf/T0

Now, we need to convert the temperatures from Celsius to Kelvin:

T0 = 18.0°C + 273 = 291 K

Tf = 35.0°C + 273 = 308 K

Finally, we can plug the values into the equation:

Pf = P0 * Tf/T0 = 7.00 x 10^5 Pa * 308 K / 291 K = 7.41 x 10^5 Pa

So, the final pressure is approximately 7.41 x 10^5 Pa. This is about a 6% increase in pressure due to the temperature change.

The ideal gas law can also be expressed in terms of moles:

PV = nRT

Where n is the number of moles and R is the universal gas constant. This form of the equation can be used in a similar way to calculate pressure changes due to temperature changes when the number of moles is known.

The ideal gas law is a useful tool for understanding and predicting the behaviour of gases, particularly when dealing with changes in pressure, volume, temperature, or the number of molecules or moles. It is important to note that the ideal gas law assumes ideal conditions, where gases behave as point masses with no intermolecular forces and negligible volume. In reality, gases may deviate from ideal behaviour at high pressures and low temperatures.

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The ideal gas law and energy

The ideal gas law, also known as the general gas equation, is a hypothetical equation of state of an ideal gas. It is a good approximation of the behaviour of many gases under many conditions. The ideal gas law is PV=nRT, where P is the absolute pressure of the gas, V is the volume of the gas, T is the absolute temperature of the gas, n is the amount of substance of gas (or number of moles), and R is the ideal or universal gas constant.

The ideal gas law is closely related to energy. The units on both sides of the equation are joules. The right-hand side of the equation, nRT, is roughly the amount of translational kinetic energy of n atoms or molecules at an absolute temperature T. The left-hand side, PV, also has the units of joules, as pressure multiplied by volume is energy. The energy of a gas is related to both its pressure and its volume. This energy can be changed when the gas is doing work as it expands, similar to what occurs in gasoline or steam engines and turbines.

The ideal gas law can be used to calculate pressure change, temperature change, volume change, or the number of molecules or moles in a given volume. It can also be used when the amount of gas is given and the mass of the gas is constant.

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When the ideal gas law is applicable

The ideal gas law is a good approximation of the behaviour of many gases under many conditions. It is often written in the form:

> PV = nRT

Where:

P is the absolute pressure of the gas
V is the volume of the gas
N is the amount of substance of gas (also known as the number of moles)
R is the ideal, or universal, gas constant
T is the absolute temperature of the gas

The ideal gas law applies perfectly when pressure approaches 0. It is a good approximation when pressure is low and temperature is high. The law does not apply to liquids.

The ideal gas law is derived from a model and, like any model, it applies where its underlying assumptions are good approximations of reality. Important assumptions for the ideal gas law include:

Point particles: In the ideal gas, particles occupy no volume. A real gas in which atoms or molecules occupy a vanishing fraction of the volume is a good approximation. This suggests low density.
Non-interacting: There are no inter-particle forces at mean distance. This suggests moderate temperatures. Mono-atomic gases will be a good approximation at higher temperatures than more complex molecules.
Random motion: The particles are in rapid, constant, random motion.

The ideal gas law can be used to solve for unknown variables when the amount of gas is given and the mass of the gas is constant.

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The ideal gas law and the combined gas law

The ideal gas law is a combination of all simple gas laws, such as Boyle's Law, Charles' Law, and Avogadro's Law. It is used to determine the pressure, volume, or temperature of a gas at certain conditions. The ideal gas law is:

PV = nRT

Where:

P = Pressure
V = Volume
N = Number of moles of gas
T = Temperature
R = Universal gas constant

The ideal gas law assumes that the gas particles have no forces acting among them and do not take up any space.

The combined gas law combines Boyle's Law, Charles' Law, and Gay-Lussac's Law. It states that the ratio of the product of pressure and volume and the absolute temperature of a gas is equal to a constant. The combined gas law formula is:

P1V1 / T1 = P2V2 / T2

Where:

P = Pressure
V = Volume
T = Absolute temperature (Kelvin)
K = Constant

The combined gas law is used when the amount of gas remains constant, and only the pressure, temperature, or volume is varied.

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The ideal gas law in thermodynamic processes

The ideal gas law is a good approximation of the behaviour of many gases under many conditions. It can be used to calculate pressure change, temperature change, volume change, or the number of molecules or moles in a given volume.

The ideal gas law is often written in the form:

PV = nRT

Where:

P is the absolute pressure of the gas
V is the volume of the gas
N is the amount of substance of gas (also known as the number of moles)
R is the ideal, or universal, gas constant
T is the absolute temperature of the gas

The ideal gas law can be used to solve for unknown variables. For example, if you know the initial pressure, volume, and temperature of a gas, you can use the ideal gas law to calculate the final pressure and volume after a change in temperature.

The ideal gas law is also useful in understanding the thermodynamic processes that gases undergo. These include:

Adiabatic processes, where heat flow (Q) is zero
Isobaric processes, where the pressure of the gas remains constant
Isochoric processes, where the volume of the gas remains constant
Isothermal processes, where the temperature of the gas remains constant

For example, in an isobaric process, because pressure is constant, the PV diagram shows a horizontal line. Applying this to the ideal gas law, you find that V/T must remain constant for the process.

In summary, the ideal gas law is a versatile tool that can be used to calculate various properties of gases and understand the thermodynamic processes they undergo.