Solving Radical Equations, 3

Category: Algebra

"Published in Newark, California, USA"

Find the roots of the equation for

Solution:

Consider the given equation above

Square both sides of the equation to remove the radical sign at the right side of the equation, we have

Equate each factor to zero in order to solve for the roots. 

If 

then

If 

then

Next, we need to check and find out if the roots are real or extraneous by substituting the values of x to the given equation as follows:

If 

then

Since both sides of the equation are equal, then x = -1 is a root of the equation.

If

then

Since both sides of the equation are also equal, then x = -2 is also a root of the equation.

Therefore, the roots of the equation are -1 and -2. 

Chemical Equilibrium

Category: Chemical Engineering Math, Algebra

"Published in Newark, California, USA"

For the reaction

at 500°C, K_eq = 62.5. If 5.0 moles of H₂ and 5.0 moles of I₂ are placed in a 10L container at 500°C and allowed to come to equilibrium, calculate the final concentration of H₂, I₂, and HI.

Solution:

Consider the given chemical reaction above

                Start                   5.0           5.0           0
                End                 5.0 - x       5.0 - x        2x

Let x be the number of moles of H₂ that will react with I₂ to form HI. Since the coefficient of H₂ and I₂ are the same, then the number of moles of I₂ that will react with H₂ to form HI is also x. 

Let 2x be the number of moles of HI that is formed by the reaction of H₂ and I₂. Because of the stoichiometric relationship of H₂, I₂, and HI, then x is the number of moles of H₂, x is the number of moles of I₂, and 2x is the number of moles of HI.

Note: It is important to balance the chemical reactions first all the time in order to assign the number of moles of the reactants and products correctly either at the start or at the end of the reaction.

Since the given chemical reaction is equilibrium or reversible reaction, then the equilibrium constant is calculated as follows

where [HI], [H₂], and [I₂] are the concentration of reactants and products in molarity or moles of solute per liter of solution. 

Calculate the concentration of [HI], [H2], and [I2] as follows:

Substitute the above values to the above equation, we have

Take the square root on both sides of the equation, we have

Therefore, the final concentration of the product:

Therefore, the final concentration of the reactants:

Inspecting Integrating Factors

Category: Differential Equations, Integral Calculus, Algebra

"Published in Newark, California, USA"

Find the general solution for

Solution:

Consider the given equation above

Expand the above equation and regroup as follows

Divide both sides of the equation by y², we have

The grouped terms in parenthesis can be simplified as an exact differential as follows

Integrate on both sides of the equation, we have

Multiply both sides of the equation by y, we have

Therefore, the final answer is