Category: Chemical Engineering Math, Differential Equations
"Published in Vacaville, California, USA"
The half-life of Sr-90 is 29 years. What fraction of the atoms in a sample of Sr-90 would remain in 100 years later?
Solution:
From the description of a given problem, it is about exponential decay problem. The rate of change of a substance is directly proportional to the negative of its substance present. The working equation can be expressed as follows
where
x = amount of Sr-90 at time t
t = decaying time of Sr-90
k = proportional constant for decaying
By separation of variables, transfer x to the left side of the equation and dt to the right side of the equation as follows
Integrate on both sides of the equation, we have
Take the inverse natural logarithm on both sides of the equation, we have
If x = x0 and t = 0 at the start, then the value of C is
Hence, the particular solution of the working equation is
If x = ½ x0 at t = 29 years, then the value of k is
Take natural logarithm on both sides of the equation, we have
Hence, the complete working equation of the above equation is
If t = 100 years, therefore, the fraction of Sr-90 in a sample is
Category: Chemical Engineering Math
"Published in Newark, California, USA"
Ra-226 decays by alpha emissions. What is its decay product?
Solution:
The given word problem is about nuclear chemistry. Nuclear chemistry is the subfield of chemistry dealing with radioactivity, nuclear processes, such as nuclear transmutation, and nuclear properties. In this type of process, an element can be converted into another kind of element because its atomic number as well as its atomic weight will be changed. There are three types of emissions: alpha emission, beta emission, and gamma emission.
Ra-226 decays by alpha emissions as follows
where X is the unknown element. The atomic number and atomic weight of an unknown element is.
By looking the Periodic Table of Elements, the unknown element is found out to be Radon which is located at Noble Gases.
Hence, the above nuclear equation can be written as follows
Therefore, the decay product Ra-226 is Rn-222. Alpha can also be written as He.
Category: Chemical Engineering Math
"Published in Newark, California, USA"
The osmotic pressure of a dilute aqueous ethyl alcohol solution at 25°C is 38 mm Hg. Calculate the boiling point elevation of this solution.
Solution:
Osmotic pressure is the minimum pressure which needs to be applied to a solution to prevent the inward flow of water across a semipermeable membrane. It is also defined as the measure of the tendency of a solution to take in water by osmosis.
The osmotic pressure of any solution is given by the equation
where
Π = is the osmotic pressure of a solution
M = is the molarity of a solution
R = is the universal gas law constant
T = is the absolute temperature of a solution
The absolute temperature of a solution is
The universal gas law constant for grams, gmole, mm Hg, liters, and K is 
.
Hence, the molarity of ethyl alcohol solution is
The boiling point elevation is defined as the product of the boiling
point constant of a solvent and the molality of a solution.
where m is the molality of a solution.
From the Table of Freezing and Boiling Information of Solvents, the boiling point constant of water is 
.
Since the concentration of ethyl alcohol is in molarity, then we have to convert it into molality as follows
where ρ is the mass density of a solution.
By looking at the Table of Ethyl Alcohol - Water Mixtures, the density in kg/L is almost the same or similar to pure water which has a range from 0.9 to 1.0 kg/L although we increase the concentration of ethyl alcohol from 0 to 50% by weight. In this case, let's approximate that the density of ethyl alcohol solution is 1.00 kg/L.
Hence, the molality of ethyl alcohol solution is
The unit of molality must be gmoles of ethyl alcohol per kg. of water. The weight of ethyl alcohol in 1 liter of solution is
The weight of water in 1 liter of solution is
Since the weight of water is almost equal to the weight of solution, then the molality of ethyl alcohol is equal to
Therefore, the boiling point elevation of ethyl alcohol solution is
