Created in the early 17th century, the gas laws have been around to assist scientists in finding volumes, amount, pressures and temperature when coming to matters of gas. The gas laws consist of three primary laws: Charles' Law, Boyle's Law and Avogadro's Law (all of which will later combine into the General Gas Equation and Ideal Gas Law). The three fundamental gas laws discover the relationship of pressure, temperature, volume and amount of gas. Boyle's Law tells us that the volume of gas increases as the pressure decreases. Charles' Law tells us that the volume of gas increases as the temperature increases. And Avogadro's Law tell us that the volume of gas increases as the amount of gas increases. The ideal gas law is the combination of the three simple gas laws. Ideal gas, or perfect gas, is the theoretical substance that helps establish the relationship of four gas variables, pressure (P), volume(V), the amount of gas(n)and temperature(T). It has characters described as follow:
Real gas, in contrast, has real volume and the collision of the particles is not elastic, because there are attractive forces between particles. As a result, the volume of real gas is much larger than of the ideal gas, and the pressure of real gas is lower than of ideal gas. All real gases tend to perform ideal gas behavior at low pressure and relatively high temperature. The compressiblity factor (Z) tells us how much the real gases differ from ideal gas behavior. \[ Z = \dfrac{PV}{nRT} \] For ideal gases, \( Z = 1 \). For real gases, \( Z\neq 1 \).
In 1662, Robert Boyle discovered the correlation between Pressure (P)and Volume (V) (assuming Temperature(T) and Amount of Gas(n) remain constant): \[ P\propto \dfrac{1}{V} \rightarrow PV=x \] where x is a constant depending on amount of gas at a given temperature.
Another form of the equation (assuming there are 2 sets of conditions, and setting both constants to eachother) that might help solve problems is: \[ P_1V_1 = x = P_2V_2 \]
In 1787, French physicists Jacques Charles, discovered the correlation between Temperature(T) and Volume(V) (assuming Pressure (P) and Amount of Gas(n) remain constant): \[ V \propto T \rightarrow V=yT \] where y is a constant depending on amount of gas and pressure. Volume is directly proportional to Temperature
Another form of the equation (assuming there are 2 sets of conditions, and setting both constants to eachother) that might help solve problems is: \[ \dfrac{V_1}{T_1} = y = \dfrac{V_2}{T_2} \]
In 1811, Amedeo Avogadro fixed Gay-Lussac's issue in finding the correlation between the Amount of gas(n) and Volume(V) (assuming Temperature(T) and Pressure(P) remain constant): \[ V \propto n \rightarrow V = zn\] where z is a constant depending on Pressure and Temperature.
Another form of the equation (assuming there are 2 sets of conditions, and setting both constants to eachother) that might help solve problems is: \[ \dfrac{P_1}{n_1} = z= \dfrac{P_2}{n_2}\]
The ideal gas law is the combination of the three simple gas laws. By setting all three laws directly or inversely proportional to Volume, you get: \[ V \propto \dfrac{nT}{P}\] Next replacing the directly proportional to sign with a constant(R) you get: \[ V = \dfrac{RnT}{P}\] And finally get the equation: \[ PV = nRT \] where P= the absolute pressure of ideal gas
Here, R is the called the gas constant. The value of R is determined by experimental results. Its numerical value changes with units. R = gas constant = 8.3145 Joules · mol-1 · K-1 (SI Unit)
You can get the numerical value of gas constant, R, from the ideal gas equation, PV=nRT. At standard temperature and pressure, where temperature is 0 oC, or 273.15 K, pressure is at 1 atm, and with a volume of 22.4140L, \[ R= \frac{PV}{RT} \] \[ \frac{1 atm \centerdot 22.4140L}{1 mol \centerdot 273.15K} \] \[ =0.082057 \; L \centerdot atm \centerdot mol^{-1} K^{-1} \] \[ R= \frac{PV}{RT} \] \[ = \frac{1 atm \centerdot 2.24140 \centerdot 10^{-2}m^3}{1 mol \centerdot 273.15K} \] \[ = 8.3145\; m^3\; Pa \centerdot mol^{-1} \centerdot K^{-1} \]
In an Ideal Gas situation, \( \frac{PV}{nRT} = 1 \) (assuming all gases are "ideal" or perfect). In cases where \( \frac{PV}{nRT} \neq 1 \) or if there are multiple sets of conditions (Pressure(P), Volume(V), number of gas(n), and Temperature(T)), use the General Gas Equation: Assuming 2 set of conditions: Initial Case: Final Case: \[ P_iV_i = n_iRT_i \; \; \; \; \; \; P_fV_f = n_fRT_f \] Setting both sides to R (which is a constant with the same value in each case), one gets: \[ R= \dfrac{P_iV_i}{n_iT_i} \; \; \; \; \; \; R= \dfrac{P_fV_f}{n_fT_f} \] If one substitutes one R for the other, one will get the final equation and the General Gas Equation: \[ \dfrac{P_iV_i}{n_iT_i} = \dfrac{P_fV_f}{n_fT_f} \]
If in any of the laws, a variable is not give, assume that it is given. For constant temperature, pressure and amount:
T(K) = T(oC) + 273.15 (unit of the temperature must be Kelvin) 2. Pressure: 1 Atmosphere (760 mmHg)3. Amount: 1 mol = 22.4 Liter of gas 4. In the Ideal Gas Law, the gas constant R = 8.3145 Joules · mol-1 · K-1
Dutch physicist Johannes Van Der Waals developed an equation for describing the deviation of real gases from the ideal gas. There are two correction terms added into the ideal gas equation. They are \( 1 +a\frac{n^2}{V^2}\), and \( 1/(V-nb) \). Since the attractive forces between molecules do exist in real gases, the pressure of real gases is actually lower than of the ideal gas equation. This condition is considered in the van der waals equation. Therefore, the correction term \( 1 +a\frac{n^2}{V^2} \) corrects the pressure of real gas for the effect of attractive forces between gas molecules. Similarly, because gas molecules have volume, the volume of real gas is much larger than of the ideal gas, the correction term \(1 -nb \) is used for correcting the volume filled by gas molecules. Practice Problems
1. 2.40L To solve this question you need to use Boyle's Law: \[ P_1V_1 = P_2V_2 \] Keeping the key variables in mind, temperature and the amount of gas is constant and therefore can be put aside, the only ones necessary are:
Plugging these values into the equation you get: V2=(1.43atm x 4 L)/(2.39atm) = 2.38 L 2. 184.89 K To solve this question you need to use Charles's Law: Once again keep the key variables in mind. The pressure remained constant and since the amount of gas is not mentioned, we assume it remains constant. Otherwise the key variables are:
Since we need to solve for the final temperature you can rearrange Charles's:
Once you plug in the numbers, you get: T2=(308.15 K x .75 L)/(1.25 L) = 184.89 K 3. 1000 mL or 1L Using Avogadro's Law to solve this problem, you can switch the equation into \( V_2=\frac{n_1\centerdot V_2}{n_2} \). However, you need to convert grams of Helium gas into moles. \[ n_1 = \frac{4.00g}{4.00g/mol} = \text{1 mol} \] Similarily, n2=2 mol \[ V_2=\frac{n_2 \centerdot V_2}{n_1}\] \[ =\frac{2 mol \centerdot 500mL}{1 mol}\] \[ = \text{1000 mL or 1L } \] References
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