__Category__: Chemical Engineering Math

"Published in Newark, California, USA"
How many grams of KClO_{3} are needed to prepare 1.8 L of O_{2} gas that is collected over H_{2}O at 22°C and 760 torr? Vapor pressure of water at 22°C is 19.8 torr.

__Solution__:
From the given word problem, it is about Dalton's Law of Partial Pressure because it involves the mixture of gas vapors. In this problem, oxygen gas that is collected from the decomposition of KClO_{3 }over water is not a pure oxygen. It is a mixture of pure oxygen and water vapor. By Dalton's Law of Partial Pressure, the pressure of pure oxygen is
The absolute temperature of oxygen gas is
The universal gas law constant for gmole, K, and mm Hg (torr) is .

By Ideal Gas Law, we can calculate the number of moles of oxygen gas as follows
From the decomposition of KClO_{3} by heating,
Therefore, the amount of KClO_{3} is

__Category__: Chemical Engineering Math

"Published in Newark, California, USA"
What is the volume of 18.0 grams of pure water at 4°C and 1 atm?
__Solution__:
From the given word problem, it is about Ideal Gas Law because the weight, temperature, and pressure of pure water are given and we need to solve for the volume.
The Ideal Gas Law is given by the equation
where:
P = pressure of a gas or vapor
V = volume of a gas or vapor
n = moles of a gas or vapor
T = absolute temperature of a gas or vapor
n = universal gas law constant
The number of moles of pure water is
The absolute temperature of pure water is
The universal gas law constant for gmole, K, and atmosphere is .
Therefore, the volume of pure water is

__Category__: Differential Equations

"Published in Newark, California, USA"
Find the particular solution for
when x = x_{0}; v = v_{0}.
__Solution__:
Consider the given equation above
By
separation of variables, transfer dx to the right side of the equation as follows
Integrate on both sides of the equation, we have
Substitute the values of v and x in order to solve for C as follows
Therefore, the particular solution is

__Category__: Differential Equations

"Published in Newark, California, USA"
Find the particular solution for
when Î¸ = 0, r = a.
__Solution__:
Consider the given equation above
By
separation of variables, transfer r³ to the left side of the equation as follows
Integrate on both sides of the equation, we have

Substitute the values of r and Î¸ in order to solve for C as follows

Therefore, the particular solution is
Take the inverse natural logarithm on both sides of the equation, we have