Balance the chemical equation by assigning variables to the coefficients and setting up equations for each element.
Solve the system of equations to find the smallest whole number values for the coefficients.
Substitute the values of the coefficients into the equation to obtain the balanced equation: 1 M g 3 P 2 + 3 A g 2 S → 3 M g S + 2 A g 3 P .
Complete the table based on the balanced equation to show the number of atoms of each element on both sides.
The balanced equation is: 1 M g 3 P 2 + 3 A g 2 S → 3 M g S + 2 A g 3 P
Explanation
Understanding the Problem We are given the unbalanced chemical equation:
□ M g 3 P 2 + □ A g 2 S → □ M g S + □ A g 3 P
Our goal is to find the smallest whole number coefficients that balance the equation and to complete the table showing the number of atoms of each element on both sides of the balanced equation.
Setting up Equations To balance the equation, we need to find the appropriate coefficients for each compound such that the number of atoms of each element is the same on both sides of the equation. Let's represent the coefficients as follows:
x M g 3 P 2 + y A g 2 S → z M g S + w A g 3 P
This gives us the following equations for each element:
Mg: 3x = z
P: 2x = w
Ag: 2y = 3w
S: y = z
Solving the Equations Now we need to solve this system of equations. From the magnesium equation, we have z = 3 x . From the sulfur equation, we have y = z , so y = 3 x . From the silver equation, we have 2 y = 3 w , so 2 ( 3 x ) = 3 w , which simplifies to 6 x = 3 w , or w = 2 x . From the phosphorus equation, we have w = 2 x , which is consistent with the previous result.
Finding the Coefficients To find the smallest whole number coefficients, we can choose the smallest whole number for x , which is 1. Then we have:
x = 1
y = 3x = 3(1) = 3
z = 3x = 3(1) = 3
w = 2x = 2(1) = 2
So the balanced equation is:
1 M g 3 P 2 + 3 A g 2 S → 3 M g S + 2 A g 3 P
Completing the Table Now we can complete the table based on the balanced equation:
Element
Reactant
Products
Mg
3
3
P
2
2
Ag
6
6
S
3
3
Final Answer The balanced equation is:
1 M g 3 P 2 + 3 A g 2 S → 3 M g S + 2 A g 3 P
The completed table is:
Element
Reactant
Products
Mg
3
3
P
2
2
Ag
6
6
S
3
3
Examples
Balancing chemical equations is essential in various fields, such as chemistry, environmental science, and materials science. For instance, in designing a new battery, chemists need to ensure that the chemical reactions inside the battery are balanced to achieve optimal performance and longevity. Similarly, in environmental science, balancing chemical equations helps in understanding and mitigating pollution by accurately predicting the products and their quantities in chemical reactions occurring in the environment. In materials science, it aids in synthesizing new materials with desired properties by controlling the stoichiometry of the reactants.