Ideal Gas Law - Differential Approximation due to variable increase in %.

In summary, the ideal gas law states that the pressure, temperature, and volume of a confined gas are related by P=kT/V, where K is a constant. Using differentials, we can approximate the percentage change in pressure if the temperature is increased by 3% and the volume is increased by 5%. In the original attempt at the solution, the answer was found to be 2%, but a solution found online suggests that the correct answer is 4%. This discrepancy may be related to a potential error in the original solution, where it appears that the solution divides by 1/2 without a clear reason. The solution found online may provide a clearer explanation for the problem.
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Homework Statement


This is a problem form my calculus book, which states:

According to the ideal gas law, the pressure, temperature, and volume of a confined gas are related by P=kT/V, where K is a constant. Use differentials to approximate the percentage change in pressure if the temperature of a gas is increased 3% and the volume is increased 5%.

The Attempt at a Solution



Here, I include my original attempt at the solution and a solution that I found online. In my original solution, I found the answer to be 2%, but the solution says 4%. I'm not sure why... but I think it's related to something I don't understand in the actual solution, where it seems to me, they divide by 1/2 for no apparent logical reason.

Could someone please explain to me where my logic is flawed?
attachment.php?attachmentid=40470&d=1319929174.jpg


Here is a solution I found online to the same problem, perhaps it's more clear.
attachment.php?attachmentid=40468&d=1319927946.jpg
 

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Edited to make images easier to see. (Sorry.)
 

1. What is the Ideal Gas Law?

The Ideal Gas Law is a fundamental equation in thermodynamics that describes the relationship between the pressure, volume, temperature, and number of moles of an ideal gas. It is written as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the gas constant, and T is temperature.

2. What is Differential Approximation?

Differential Approximation is a mathematical technique used to approximate the behavior of a variable when it is changing at a constant rate. In the context of the Ideal Gas Law, it is used to approximate the change in pressure, volume, or temperature when the other variables are increasing at a constant percentage.

3. How is Differential Approximation applied to the Ideal Gas Law?

In the Ideal Gas Law, Differential Approximation is used to calculate the change in pressure, volume, or temperature when there is a constant percentage increase in one of the other variables. This is done by taking the derivative of the Ideal Gas Law equation with respect to the variable that is changing, and then using that derivative to calculate the change in the other variable. This approximation is useful in situations where the exact change in a variable cannot be directly measured.

4. What is the significance of variable increase in % in the Ideal Gas Law?

The variable increase in % in the Ideal Gas Law refers to the change in one of the variables by a certain percentage. This is important because it allows us to understand how the other variables will change in response to this percentage change. This is particularly useful in practical applications, such as in the design of gas containers or in the study of gas behavior in industrial processes.

5. What are the limitations of using Differential Approximation in the Ideal Gas Law?

While Differential Approximation can provide useful approximations in many situations, it is important to note that it is not always accurate. The Ideal Gas Law only applies to ideal gases, which do not exist in the real world. Additionally, the approximation may become less accurate as the percentage change in the variable increases. Therefore, it is important to use caution when applying this technique and to consider the limitations and assumptions of the Ideal Gas Law.

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