The Combined Gas Law

In an inverse proportion the variables go in opposite ways One goes up and the other goes down. 

The multiplier is a ratio of pressures. Should this ratio be more than 1 ( ) or 

The ratio should be less than 1, so the lower pressure is on top. 

The fraction is somewhat smaller than 1, and 4.86 is somewhat smaller than 

5 For a fixed quantity of a confined gas, given the initial volume, pressure, and temperature and the final values of any two variables, calculate the final value of the third variable. 

Let us see how this combined gas law can be used in a practical situation. Example P. 

A sample of gas has a volume of 5 dm at atmospheric pressure and a temperature of 25 C. WTiat is its volume at 0 °C (Remember, atmospheric pressure and 25 C are the standard temperature and pressure, or STP.) 

Answer In this case, the unknown of the equation is the second volume, so we will rearrange the equation to put that on the left-hand side the pressure is constant throughout, so Pi — P - So  

A gas has a volume of 250 cm- at STP. What will its pressure become if the volume is changed to 600 cm In this question, the temperature is constant throughout, and the second pressure is the unknown. The two volumes are 0.25 and 0.6 dm and the first pressure is 1 atm. It may be easier to see this if we arrange the figures as in the table below  

In this question the units are given in atmospheres, so we will answer it in the same units. The answer in standard units is given for completeness. Let s rearrange the combined gas equation to give P2 on the LHS and put these figures in. 

STEP0 Substitute values into the gas law equation and calculate. When we substitute in the values, we see that the ratio of the temperatures (temperature factor), is greater than 1, which increases pressure as predicted in Step 1. 

Because the calculated pressure of 9.1 atm of the gas exceeds the limit of 8.0 atm for the can, we would expect the aerosol can to explode. 

In a storage area where the temperature has reached 55 °C, the pressure of oxygen gas in a 15.0-L steel cylinder is 965 torr. To what temperature, in degrees Celsius, would the gas have to be cooled to reduce the pressure to 850. torr when the volume and amount of the gas do not change  

LEARNING GOAL Use the temperature-pressure relationship (Gay-Lussac s law) to determine the final temperature or pressure when the volume and amount of gas are constant. 

29 Calculate the final pressure, in torr, for each of the following, if n and V are constant  

It would be equally correct to consider that first the temperature of the gas was changed from Pi to P2 at the constant pressure Pi, for which a new volume Vnew could be calculated using Charles law. Then, assuming that the temperature is held constant at P2, calculate how the volume changes as the pressure is changed from Pi to P2 (boxes 1, 2, and 4 in Fig. 12-8). 

However, the fact that the volume V of a given mass of gas is inversely proportional to its pressure P and directly proportional to its absolute temperature P can be combined mathematically to give the single equation 

That is, for a given sample of gas, the ratio PV/T remains constant, and therefore 

Note that if the temperature is constant, Pi = P2, then the expression reduces to the equation for Boyle s law, Pj Vi = P2V2. Alternatively, if the pressure is constant. Pi = P2, the expression is equivalent to Charles  

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EXAMPLE 4.4 Sample exercise Using the combined gas law when one variable is changed... 

According to the combined gas law, the volume of a given mass of gas can have any value, depending on its temperature and pressure. To compare the quantities of gas present in two different samples, it is useful to adopt a set of standard conditions of temperature and pressure. By universal agreement, the standard temperature is chosen as 273 K (0°C) and the standard pressure is chosen as exactly 1 atm (760torr). Together, these conditions are referred to as standard conditions or as standard temperature and pressure (STP). While there is nothing special about STP, some authors and some instructors find it convenient to use this short notation for this particular temperature and pressure. 

Arts. Boyle s law may be used because the temperature is unchanged. Alternately, the combined gas law may be used, with the Kelvin equivalent of 18°C used for both T[ and 7. ... 

Write an equation for the combined gas law using temperature in Celsius. Explain why the Kelvin scale is more convenient. 

When applying Boyle s Law, we are working at constant temperature and with a constant number of moles of gas. In this case, T = T2, and the Combined Gas Law simplifies to... 

The law we need here that is applied to situations where the number of moles of gas and the volume stay constant can be derived from the Combined Gas Law ... 

Plan (1) Use the Combined Gas Law to calculate the volume that 1 L of the gas would have occupied... 

C is a temperature higher than STP. This condition increases the 1.00-L volume that contains 0.44 g of the mixture at STP. We calculate the expanded volume with the combined gas law. 

We will be solving the combined gas equation for V2, so we will take the combined gas law and rearrange for V2 ... 

It is possible to combine Avogadro s law and the combined gas law to produce the ideal gas equation, which incorporates the pressure, volume, temperature, and amount relationships of a gas. The ideal gas equation has the form of... 

We can use the gas law relationships, especially the ideal gas law and the combined gas law, in reaction stoichiometry problems. For example, suppose you have 2.50 g of an impure sample of KC103 and you want to determine how many grams of pure KC103 are present. You heat the mixture and the KC103 decomposes according to the equation ... 

The presence of two volumes (or two temperatures) is a very strong indication that we will need to use the combined gas law. To use this gas law we need subscripts to differentiate the different volumes and temperatures. Label one volume Vj and the other V2. It does not matter which volume we label 1 or 2 as long as we label all associated variables with the same subscript. 

The combined gas law is (PiVi/Tj) = (P2V2/T2). It is possible to simplify this equation in this problem by removing all variables not appearing in the table. The simplified combined gas law is (V/Tj) = (V2/T2), which is a form of Charles law. After simplification, we need to isolate the variable we are seeking (the one with the question mark in the table). Isolation of T2 requires manipulating the equation. There are various ways of doing this, all yielding the equation T2 = (T /Vi). We now enter the appropriate values from our table into this equation ... 

In this chapter, you learned about the properties of gases. You learned that you can use the combined gas law, the ideal gas law, or the individual gas laws to calculate certain gas quantities, such as temperature and pressure. You also learned that these equations could also be useful in reaction stoichiometry problems involving gases. You learned the postulates of the Kinetic-Molecular... 

In the discussion of Boyle s, Charles s, and Gay-Lussac s laws we held two of the four variables constant, changed the third, and looked at its effect on the fourth variable. If we keep the number of moles of gas constant—that is, no gas can get in or out—then we can combine these three gas laws into one, the combined gas law, which can be expressed as ... 

For example, suppose a 5.0-L bottle of gas with a pressure of 2.50 atm at 20°C is heated to 80°C. We can calculate the new pressure using the combined gas law. Before we start working mathematically, however, let s do some reasoning. The volume of the bottle hasn t changed, and neither has the number of moles of gas inside. Only the temperature... 

We could work this into the combined gas law, but more commonly the amount of gas is related to the other physical properties through another relationship that Avogadro developed ... 

Most gas law experiments use either the combined gas law or the ideal gas equation. Moles of gas are a major factor in many of these experiments. The combined gas law can generate the moles of a gas by adjusting the volume to STP and using Avogadro s relationship of 22.4 L/mol at STE The ideal gas equation gives moles from the relationship n = PV/RT. 

The values of P, T, and n may be used to determine the volume of a gas. If this volume is to be used with Avogadro s law of 22.4 L/mol, the combined gas law must be employed to adjust the volume to STE This equation will use the measured values for P and Talong with the calculated value of V. These values are combined with STE conditions (0°C (273.15 K) and 1.00 atm) to determine the molar volume of a gas. 

Use the combined gas law to calculate molar volume at room temperature and atmospheric pressure. 

Boyle s and Charles s laws are convenient if you happen to find yourself in situations where only two factors change at a time. But the universe is rarely so well-behaved. What if pressure, temperature, and volume all change at the same time Are aspirin and a nap the only solution No. Enter the combined gas law ... 

As a final gift, you may like the following equations. Sometimes working with the combined gas law can be a serious pain in the butt due to all the algebra involved with recirranging the equations endlessly. So here you go the combined gas law solved for every Vciriable Please use this cimazing new information spciringly. 

L. The number of moles of gas (10 mol) remains constant. The other three factors (pressure, temperature, and volume) all change between initial and final states, so you need to use the combined gas law. The initial values (290 atm, 283 K, 0.80 L) all come from the excimple problem. The final temperature and pressure are known (273 K, 1 atm) because the question states that the gas ends up at STP. So the only unknown is the final volume. Recirrange the combined gas law to solve for this value ... 

L. You re given an initial volume, initial temperature, and initial pressure. You re also given a final pressure. The problem doesn t specifically tell you what the final temperature is, but you re told the temperature increases by 10°C, so you can determine the final temperatire by adding 10 to the initial temperature. (Be sure to then convert both temperatures to kelvins. The initial temperature is 27.0°C+273 = 300 K, and the final temperature is 37°C+273 = 310 K.) The only unknown is final volume. Rearrange the combined gas law to solve for final volume, V ... 

Strategy Compression increases the pressure but cooling lowers the pressure, so it is not easy to predict the outcome without doing a calculation. Summarize the known and unknown information in a table (remember to convert temperatures to the Kelvin scale) and rearrange the combined gas law to give the unknown (in this case, the pressure) in terms of the other variables. Cancel any constant terms (n, in this case) and substitute the data. 

The density (D) of a gas varies inversely to the volume, as shown in equation (a). We can substitute a rearrangement of the combined gas law from equation (b) to provide equation (c), which describes changes in density with temperature and pressure. 

Since we have all three factors involved in gas action/reaction, we can apply the combined gas law. The 6-cylinder engine, 6.0 liters total volume, has 1.0-L cylinders, which is the value of V. The remainder of the factors are stated. 

Remembering that all gas law problems require temperature expressed in the Kelvin scale, you can apply the combined gas law to determine the answer. 

The solution for this problem is derived from the use of the combined gas law. 

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