Question

Molecules A sugar molecule has twice as many atoms of hydrogen as it does oxygen and one more atom of carbon than of oxygen. If a sugar molecule has a total of 45 atoms, how many are oxygen? How many are hydrogen?

Short answer

The sugar molecule has 11 oxygen atoms and 22 hydrogen atoms.

Step-by-step solution

Identify Variables

Let the number of oxygen atoms be denoted by O. According to the problem, the number of hydrogen atoms is twice that of oxygen, so let it be denoted by H = 2O. The number of carbon atoms is one more than the number of oxygen atoms, so let it be denoted by C = O + 1.

Set Up the Total Atoms Equation

The total number of atoms is given as 45. Thus, we set up the equation representing the total number of atoms: O + H + C = 45.

Substitute Variables

Replace H and C with their expressions in terms of O. This gives: O + 2O + (O + 1) = 45.

Simplify the Equation

Combine like terms: 4O + 1 = 45.

Solve for O

Subtract 1 from both sides: 4O = 44. Divide both sides by 4: O = 11.

Calculate Number of Hydrogen and Carbon Atoms

Using the value of O, calculate H and C: H = 2O = 2(11) = 22. C = O + 1 = 11 + 1 = 12.

Key concepts

Variable Identification

Understanding the problem starts with identifying the right variables. Let's break it down. We need to find the number of different atoms in a sugar molecule: oxygen, hydrogen, and carbon. To start, we'll designate variables for the unknown values.
Let the number of oxygen atoms be denoted by O. This is our baseline variable.
From the problem, we know that the number of hydrogen atoms is twice the number of oxygen atoms. Thus, we represent hydrogen atoms as H = 2O.
Next, the problem states that there is one more carbon atom than oxygen atoms, so we can write carbon atoms as C = O + 1.
By assigning these variables, we're setting ourselves up to solve the problem in a structured way.

Atom Quantity Relation

Next, we determine how these variables (atoms) relate to each other. We’ve already set our variables:
O: Oxygen atoms
H = 2O: Hydrogen atoms are twice the oxygen atoms
C = O + 1: Carbon atoms are one more than the oxygen atoms
It's crucial to understand these relationships because these relations will help us build our total equation. Keep in mind, these relations come directly from the problem statement.

Total Equation Setup

Now that we have our variables and relationships set, it's time to build the total equation. According to the problem, the sugar molecule has a total of 45 atoms. We sum up all atoms:
Oxygen atoms: O
Hydrogen atoms: H = 2O
Carbon atoms: C = O + 1
Adding these up gives us the equation: O + H + C = 45
Using our variable relations, we substitute for H and C:
O + 2O + (O + 1) = 45
This is the total equation representing all atoms in the molecule.

Equation Simplification

With our total equation O + 2O + (O + 1) = 45, the next step is to simplify it for easier solving. We start by combining like terms:
O + 2O + O + 1 = 45
This simplifies to 4O + 1 = 45
At this point, we have a simple linear equation that needs one more step to isolate O.
Subtracting 1 from both sides, we get:
4O = 44

Solving Algebraic Equations

In the last step, we simplify the equation further to find the value of O. We have:
4O = 44
To solve for O, we divide both sides of the equation by 4:
O = 44 / 4
O = 11
Now that we have the number of oxygen atoms, we can find the number of hydrogen and carbon atoms using the relationships established earlier:
H = 2O = 2(11) = 22
C = O + 1 = 11 + 1 = 12
Therefore, the sugar molecule contains 11 oxygen atoms, 22 hydrogen atoms, and 12 carbon atoms. Understanding each step helps you solve similar problems with ease.