How knowing units can help you remember equations tested on the MCAT

MCAT

The chemistry and physics section of the MCAT is notoriously daunting. However, while questions may seem perplexing with complicated equations and challenging calculations, one can answer questions in this part of the exam quickly by knowing units of variables commonly encountered in chemistry and physics. 

Knowing units well can help you avoid memorizing a plethora of different equations. While some equations have constants that would be impossible to derive using units alone (e.g. those that are obtained by integration like kinetic energy), some common equations can be derived through algebraic manipulation using the units of the variables in the equation.  Let’s review an example. I have a cylindrical tank with height h with a certain volume, V,  filled with liquid of density , and I want to know what the pressure, P,  is at the bottom of the tank. I may remember that P = ⍴gh. But, let’s say I can’t remember this equation. How do I figure out what the pressure is based on units? 

Take a look at the units of the entity you’re trying to find. Remember, Pressure = Force/area. I know that force has units of newtons (N). Area has SI units of m2. Therefore, Pressure = N/m2. In order for two sides of the equation to be equivalent, the units must be equivalent. So, I’m given volume, which I know has standard units of m3. I’m also given a height, which has standard units of m. Finally, I’m given density, which has SI units of kg/m3. At this point, I can ask myself: how do I arrange the right side of the equation in such a way that I obtain units of N/m2. Well, I need to obtain units of force in the numerator. A tank that’s sitting on the ground and not experiencing any acceleration other than gravity experiences a force of its mass, m, multiplied by the acceleration due to gravity, g; Force = mg, which ultimately has units of kg ∙ m/s2 = N. 

Now I have a way to obtain units of force in the numerator. That is, I’m given density and I know the acceleration of gravity, so if I multiply density by gravity, I obtain units of kg/m3 ∙ m/s2 = N/m3 because the numerator is mass multiplied by acceleration, which has units of N. The final thing to do is to transform these units into units of pressure. The pressure that I want is the pressure at the bottom of the tank. I’m given the height, h, of the tank, which has units of m. If I multiply N/m3 by height in meters, I obtain units of N/m2, which is pressure. So, our final equation is P = ⍴gh, where P is the pressure at the bottom of the tank. We obtained the desired equation just by considering the units of variables given to us in the question! 

In summary, we want to consider the following when we’re stuck and can’t remember an equation: 

  1. Identify the units of the desired answer, in this case it’s units of pressure
  2. Look at the variables given and identify the units of those variables
  3. Consider how one can multiply or divide those variables to obtain the units of the desired answer

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