The data given in the table describes the behaviour of a sample of gas. Which of the following empirical laws does the data illustrate? 𝑘 is a constant. A) 𝑉 equals 𝑘𝑇 at constant 𝑝. B) 𝑝𝑇 equals 𝑝 one plus 𝑝 two plus 𝑝 three and so on at constant 𝑉 and 𝑇. C) 𝑝 equals 𝑘 over 𝑉 at constant 𝑇. D) 𝑝 equals 𝑘𝑇 at constant 𝑉. Or E) 𝑝 equals 𝑘𝑛 at constant 𝑉 and 𝑇 and 𝑛 is the amount in moles.
𝑝, 𝑉, and 𝑇 refer, respectively, to pressure, volume, and temperature. We’ve been told that this data relates to a sample of gas. We can imagine the gas in a container where we’re measuring the pressure and changing the temperature, although we’re measuring the temperature in kelvin and we’re going down to 25 kelvin. This would be extremely cold, require very specialist equipment to achieve. We’ve also got quite a large reactor at 100 litres, which is about what you’d expect for a battery actor in industry. On the other hand, our pressures aren’t that ridiculous. We’re going from room pressure at one atmosphere to half an atmosphere in one direction and only four atmospheres in the other. Some industrial process go up to much, much higher pressures. For instance, in the harbour process, we’re looking at hundreds of atmospheres.
Thankfully, we’re not being asked to work out how to actually do this in the lab. All we need to do is find the empirical law that accounts for the data. If something’s empirical, it means it’s based on experiments and real-world data. Empirical laws just described the way the data behaves. It’s not based on theory. So some of these equations may look familiar. But we’re only looking for the equation that describes the data we have. Let’s start with volume is equal to a constant multiplied by the temperature. The first thing you might notice about the data is that the volume is always constant. And it’s the pressure and temperature that are changing. So the first problem with this is that we’re not dealing with a constant pressure situation.
So this law can’t apply. But even if it did, there’s another problem. 𝑘 is supposed to be a constant. So volume and temperature should be directly proportional. But for the first data point, where we have a volume of 100 and a temperature of 200, volume divided by temperature equals 0.5. But for the second data point, it’s equal to one and then two and then four. So our constant isn’t really a constant. So this isn’t the correct answer. Now, a word of warning, the relationship that volume is directly proportional to temperature at constant pressure is called Charles’s law. It is a valid law. But unfortunately, we don’t have the data here to prove it.
In this data, the volume is constant. So we can’t look at volume–temperature relationships. So we’re still on the hunt for our empirical law. In our next law, pressure and temperature are said to be equal to the sum of individual pressures. However, the law also stipulates that we’re dealing with constant volume and temperature conditions. While the volume is constant, that temperature is not. We could also disprove this one by dimensional analysis. On the left, we have pressure times temperature, which means our units will be atmosphere kelvin, while on the right we only have pressure terms. So they sum to a value in atmospheres. So not only can it not describe our data, but it’s not a valid equation.
In the next option, pressure is equal to some constant divided by the volume. However, it also stipulates with dealing with constant temperature conditions, which we’re not. Just like with the first option, we can test by multiplying that pressure by the volume. And we end up with values of 400, 200, and so on. So we don’t have a constant 𝑘. This is, however, a valid relationship if the temperature is constant. And it’s known as Boyle’s law. The next expression is that pressure is equal to some constant multiplied by the temperature. And it stipulates that the volume is constant, which matches our data. We can check that the equation makes sense by working out the value for 𝑘 for all our data points.
For simplicity, I’m going to emit the units. Our pressure, four, divided by our temperature, 200, is equal to one over 50. Two divided by 100 is also one over 50. And the value for our constant is one over 50 for the other two data points. So this particular empirical law accurately describes the relationships in our data with a value of 𝑘 of 0.2 atmospheres per kelvin. But just to be safe, let’s have a look at the last one anyway. The law here says that pressure is equal to some constant multiplied by the amount of gas in moles. But it also stipulates that the temperature is constant, which doesn’t match our data.
Also 𝑛, the amount in moles, is not a part of our data set. So we can’t use it in any of our empirical laws. Theoretically, we could calculate 𝑛 using the ideal gas law. But since it’s not actually part of our measurement, that will be moving away from empirical principles. This example is actually consistent with the ideal gas law. Where 𝑅, 𝑇, and 𝑉 are all constant, we can summarise them into one constant term. However, it doesn’t empirically describe the data we have. So it’s not a correct answer, meaning of the five laws given the only one that adequately illustrates the data we have is 𝑝 equals 𝑘𝑇 at constant 𝑉.