Resultant of Forces and Components

Category: Mechanics, Physics

"Published in Newark, California, USA"

Determine the x and y components of each of the forces shown in the figure. Also, determine the resultant of the forces and its direction.

Photo by Math Principles in Everyday Life

Solution:

The first thing that we need to do is to draw the rectangular component of the given forces and then isolate into x and y components as follows

Photo by Math Principles in Everyday Life

By using trigonometric functions of right triangles, we can solve for the x and y components of the three given forces.

For F = 390 lbs:

For P = 300 lbs:

 

 

 

 

 

For T = 400 lbs:
 

 

 

 

 

 

         
The sum of the x components of the three given forces is
 

 

The sum of the y components of the three given forces is
 

 

 

Since the sign of R_x and R_y are both negative, then they are located at the 3rd quadrant as shown in the figure 

Photo by Math Principles in Everyday Life

By Pythagorean Theorem, the value of the resultant of the given three forces is

The direction of the resultant is

 

 

 

                               or

Ionization Constant for Acid Problems, 4

Category: Chemical Engineering Math

"Published in Newark, California, USA"

In a salt manufacturing firm in California, there's a spill of 10 gallons of concentrated hydrochloric acid (37.5% HCl solution) from the gate valve of 1,000 gallon tank that contains concentrated hydrochloric acid. Before the disposal of the spill, the state law requires to neutralize the hazardous waste first by adding sodium bicarbonate and then wipe-off the neutralized spill. How many bags of sodium bicarbonate will be needed for neutralization process? One bag of sodium bicarbonate weighs 50 pounds. Is it safe to dispose the blankets used to wipe-off the neutralized spill? What is the pH of the neutralized spill? The pH range required for the disposal of industrial waste in California is 2.0 to 12.5. Outside of the range is considered as a corrosive. (density of concentrated hydrochloric acid = 1.19 g/mL; density of sodium bicarbonate = 2.20 g/mL; molecular weight of hydrochloric acid = 36 g/mole;  and molecular weight of sodium bicarbonate = 84 g/mole)

Solution:

Since the spill is a strong acid, then the state law of California requires to neutralize it with a weak base. We cannot use any strong base because they are hazardous and corrosive, too. Same thing with a strong base. We need to neutralize it with a weak acid. That's the law for the safety of everyone and the environment as well. 

The weight of concentrated hydrochloric acid which is also 37.5% HCl solution is

The weight of pure hydrochloric acid in 37.5% HCl solution is

The number of moles of hydrochloric acid in 37.5% HCl solution is

If sodium bicarbonate is added to the spill that contains concentrated hydrochloric acid, then the chemical reaction is

                   or

The weight of sodium bicarbonate used to neutralize hydrochloric acid is 

Therefore, the number of bags of sodium bicarbonate used is

                               or

If the spill is happened in daytime especially in warmer climate, then the neutralization of concentrated hydrochloric acid with sodium bicarbonate is

Carbon dioxide is not soluble in water. When you add sodium bicarbonate to the spill which contains concentrated hydrochloric acid, then bubbles will form. Those bubbles are carbon dioxide that are escaping from the spill. 

Since sodium chloride is a neutral salt (product of strong acid and strong base), then sodium ion and chloride ion have no reaction with water. Because of this, the pH of a neutralized spill is
 

Since the pH of a neutralized spill is less than 12.5, then it is safe to dispose the blankets used to wipe-off the neutralized spill.

If the spill is happened in nighttime especially in colder climate, then the neutralization of concentrated hydrochloric acid with sodium bicarbonate is 

Since carbonic acid is a weak acid, then it is partially ionized as follows

The number of moles of carbonic acid is 

The total volume of a neutralized spill is

The concentration of carbonic acid in molarity is

From Table of Ionization Constant of Acids and Bases, carbonic acid is ionized in two steps as follows

In this neutralization process, we will use the ionization constant at the first equation.

The ionization constant of carbonic acid is

where x is the amount of carbonic acid that is converted into hydrogen carbonate ion and hydrogen ion. 
 
Since the value of K_a is less than 1 x 10^-3, then we can neglect x at the denominator as follows 

Hence, the concentration of hydrogen ion is

Therefore, the pH of a neutralized spill is

Since the pH of a neutralized spill is greater than 2.0, then it is safe to dispose the blankets used to wipe-off the neutralized spill.