Kara Sears
The ideal gas law is used to calculate pressure change, temperature change, volume change, or the number of molecules or moles in a given volume. It is applicable when a gas is at a high temperature and low pressure. The law can be used to calculate the pressure in a car tire or a bicycle tire. For example, the pressure in a car tire with a gauge pressure of 2.50 × 10^5 N/m^2 at a temperature of 35ºC can be calculated using the ideal gas law. Similarly, the pressure in a bicycle tire with an absolute pressure of 7.00 × 10^5 Pa (a gauge pressure of just under 90.0 lb/in^2) at a temperature of 18ºC can also be calculated using the ideal gas law.
| Characteristics | Values |
|---|---|
| Ideal gas law equation | PV = nRT |
| Ideal gas constant (R) | 8.314... J/mol·K |
| Boltzmann's constant (k) | 1.38 x 10^-23 J/K |
| Avogadro's number (NA) | 6.02 x 10^23 particles/mole |
| Applicable conditions | High temperature and low pressure |
| Assumptions | Molecules have negligible volume, high velocity, and minimal interaction |
| Accuracy | Inaccuracies increase at low temperatures and high pressures |
| Example | SCUBA tank at 200 bar follows the law with <3% error |
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What You'll Learn
The ideal gas law equation: PV = nRT
The ideal gas law, also called 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 various conditions. The modern form of the equation relates pressure, volume, and temperature in two main forms. The ideal gas law equation is PV = nRT, where P is pressure, V is volume, T is temperature, n is the number of moles, and R is the universal gas constant.
The ideal gas law is often used to calculate pressure change, temperature change, volume change, or the number of molecules or moles in a given volume. For example, the ideal gas law can be used to calculate the pressure in a car tyre containing a certain number of moles of gas at a specific volume and temperature. The pressure in the tyre is changing only because of changes in temperature.
The universal gas constant, R, is a constant of units of energy per temperature increment per mole. It is also known as the molar gas constant. The value of R depends on the units used in the calculation. For example, in SI units, the ideal gas law is written as PV = nRT, where P is measured in pascals (Pa), V is measured in cubic metres (m^3), n is measured in moles, and T is measured in kelvin (K).
The ideal gas law can also be used to calculate the pressure in a bicycle tyre with a certain volume, pressure, and temperature. The pressure in the bicycle tyre is initially equal to atmospheric pressure, and the volume increases in direct proportion to the number of atoms and molecules put into the tyre. Once the volume of the tyre is constant, the ideal gas law predicts that the pressure should increase in proportion to the number of atoms and molecules.
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Pressure, temperature, and volume changes
The ideal gas law is used to calculate pressure, temperature, volume, and the number of molecules or moles in a given volume. The equation for the ideal gas law is PV=nRT, where P is the pressure of the gas, V is its volume, n is the number of moles of the gas, T is its temperature in Kelvin, and R is the ideal (universal) gas constant.
The ideal gas law is closely related to energy, with the units on both sides being joules. The right-hand side of the equation, NkT, is the amount of translational kinetic energy of N atoms or molecules at an absolute temperature T. The left-hand side, PV, is the energy in a gas related to its pressure and 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 is consistent with the behaviour of filling a tire slowly when the temperature is constant. Initially, the pressure P is approximately equal to atmospheric pressure, and the volume V increases in direct proportion to the number of atoms and molecules N added to the tire. Once the volume of the tire is constant, the equation PV = NkT predicts that the pressure should increase in proportion to the number N of atoms and molecules. For example, suppose a bicycle tire is fully inflated, with an absolute pressure of 7.00 × 105 Pa (a gauge pressure of just under 90.0 lb/in2) at a temperature of 18.0°C. What will be the pressure after the temperature rises to 35.0°C, assuming no leaks or changes in volume?
Gay-Lussac's Law states that the pressure of a given amount of gas held at constant volume is directly proportional to its Kelvin temperature. Heating a gas increases the energy of its molecules, causing them to move faster and impact the walls of the container more frequently, thereby increasing the pressure. Conversely, cooling the gas slows down the molecules, reducing the number of impacts and decreasing the pressure.
Charles' Law describes the relationship between volume and temperature when pressure and the amount of gas are held constant. According to this law, if the Kelvin temperature of a gas increases, its volume also increases, and if the temperature decreases, the volume decreases.
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The universal gas constant
The value of the universal gas constant is R = NA * k, where NA is Avogadro's constant and k is the Boltzmann constant. The SI value of the universal gas constant is 8.31432 J⋅K−1⋅mol−1. This value is defined by the U.S. Standard Atmosphere, 1976, and is used by NASA, NOAA, and the USAF.
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Calculating pressure change
The ideal gas law can be used to calculate pressure changes, temperature changes, volume changes, or the number of molecules or moles in a given volume. The formula for the ideal gas law is PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature.
To calculate pressure change using the ideal gas law, you need to know the values of the other variables in the equation. For example, let's say you have a bicycle tire with a volume of 2.00 x 10^-3 m^3 (2.00 L) and a temperature of 18.0ºC. The tire is fully inflated, and you want to find the pressure after the temperature rises to 35.0ºC. Assuming there are no leaks or changes in volume, you can use the ideal gas law to calculate the pressure change.
First, convert the known values into proper SI units: volume (V) = 2.00 x 10^-3 m^3, temperature (T) = 323.15 K (18.0ºC + 273.15), and the universal gas constant (R) = 8.314 J/mol·K. Next, identify the number of moles (n) of gas in the tire. Using the ideal gas law, we can rearrange the equation to solve for n: n = PV/RT. Plugging in the values, we get n = (7.00 x 10^5 Pa x 2.00 x 10^-3 m^3) / (8.314 J/mol·K x 323.15 K), which gives us n = 0.0303 mol.
Now that we have the number of moles, we can calculate the pressure at the higher temperature. Using the ideal gas law equation PV = nRT, we can rearrange it to solve for pressure: P = nRT/V. Substituting the values, we get P = (0.0303 mol x 8.314 J/mol·K x 323.15 K) / 2.00 x 10^-3 m^3, which gives us P = 4.03 x 10^5 Pa.
Therefore, the pressure in the bicycle tire increased from 7.00 x 10^5 Pa to 4.03 x 10^5 Pa when the temperature rose from 18.0ºC to 35.0ºC. This calculation demonstrates how the ideal gas law can be used to determine pressure changes by taking into account the relationships between pressure, volume, temperature, and the number of moles of gas.
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The Z factor (compressibility factor)
The Z factor, also known as the compressibility factor or the gas deviation factor, is a measure of how closely a real gas behaves compared to an ideal gas. It is defined as the ratio of the molar volume of a gas to the molar volume of an ideal gas, both at the same temperature and pressure. The Z factor is a dimensionless number close to 1.00 and is influenced by gas gravity, gas temperature, gas pressure, and the critical properties of the gas.
The ideal gas law is reasonably accurate up to a pressure of around 2 atm, and even higher for small non-associating molecules. However, the ideal gas law does not account for the behaviour of real gases, which can deviate significantly from ideal behaviour, especially when the gas is close to a phase change, at low temperatures, or under high pressure. The Z factor is used to correct for these deviations and modify the ideal gas law to better represent real gas behaviour.
The Z factor can be calculated using equations of state (EOS) or industry correlations, with the most common approach being the use of generalized compressibility factor charts. These charts plot reduced pressure on the x-axis and Z on the y-axis, with reduced pressure defined as the ratio of absolute pressure to critical pressure. By locating the given reduced pressure on the x-axis and moving upwards until the given reduced temperature is found, the Z factor can be determined at their intersection point.
The compressibility factor is particularly important in applications such as gas pipelines, where pressure and temperature vary along the length of the pipeline, and the Z factor must be calculated for an average pressure at each location. The Standing-Katz chart is an example of a widely used compressibility factor chart.
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Frequently asked questions
The ideal gas law relates the pressure and volume of a gas to the number of gas molecules or moles of gas and the temperature of the gas.
The ideal gas law equation is PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the ideal gas constant, and T is temperature.
Yes, you can use gauge pressure in the ideal gas law equation. For example, you can calculate the pressure change of a gas using the ideal gas law, and gauge pressure is one way to measure pressure.
