Q28P - Fundamentals Of Physics

Question

A mole of atoms is $\mathbf{6}\mathbf{.}\mathbf{20}\mathbf{\times }{\mathbf{10}}^{\mathbf{23}}$atoms. To the nearest order of magnitude, how many moles of atoms are in a large domestic cat? The masses of a hydrogen atom, an oxygen atom, and a carbon atom are 1.0 u, 16 u, and 12 u, respectively. (Hint: Cats are sometimes known to kill a mole.)

Step-by-Step Solution

Answer

The number of moles in a large domestic cat is 1000.

Step 1: Given data

Number of atoms in one mole are  $6.02\times {10}^{23}$

Mass of H atom is $1.0 \mathrm{u}$

Mass of O atom is $16 \mathrm{u}$

Mass of C atom is $12 \mathrm{u}$

Step 2: Understanding the concept

This is an estimation problem. So, you need to assume the mass and the composition of the cat and find out the total number of atoms in the cat. From that, you can find the number of moles in the cat. Let’s assume the mass of the cat is 10 kg.

Step 3: Determination of the number of carbon atoms

Assume the mass of cat is 10 kg.

Assuming that $60\%$ of the cat is water and remaining $40\%$ is carbon, you can find the number of atoms of the cat.

40% of C would give you 4000 g of C.

$\frac{4000}{12}\times 6.02\times {10}^{23}=2\times {10}^{26} \mathrm{atoms} \mathrm{of} \mathrm{carbon}$

Thus, there are $2\times {10}^{26} \mathrm{atoms} \mathrm{of} \mathrm{carbon}$

Step 4: Determination of the number of oxygen atoms

In ${\mathrm{H}}_2\mathrm{O}$ there are two hydrogens and one oxygen, therefore,

 $\left(1\mathrm{u}\times 2\right)+16\mathrm{u}=18\mathrm{u}$.

You have $16 \mathrm{u}$ of oxygen in $18 \mathrm{u} {\mathrm{H}}_2\mathrm{O}$, therefore, $6000 \mathrm{g}$ of  ${\mathrm{H}}_2\mathrm{O}$ will be,

$6000 \mathrm{g} \times \frac{16\mathrm{u}}{18\mathrm{u}}=5333.33\mathrm{g}$ of oxygen.

So, the number of oxygen atoms would be,

$\frac{5333.33}{16}\times 6.02\times {10}^{23}=2\times {10}^{26}$

Thus, there are $2\times {10}^{26}$$\mathrm{atoms} \mathrm{of} \mathrm{oxygen}$.

Step 5: Determination of the number of hydrogen atoms

You have $2 \mathrm{u}$ of oxygen in $18 \mathrm{u} {\mathrm{H}}_2\mathrm{O}$, therefore, $6000\mathrm{g}$ of  ${\mathrm{H}}_2\mathrm{O}$ will have,

$6000\mathrm{g}\times \frac{2\mathrm{u}}{18\mathrm{u}}=666.667\mathrm{g}$ of hydrogen.

So, number of hydrogen atoms would be,

$\frac{666.667}{1}\times 6.02\times {10}^{23}=4\times {10}^{26}$

Thus, there are $4\times {10}^{26}$.

Step 6: Determination of the total number of atoms in the cat

The total number of atoms in the cat would be,

$\left(2\times {10}^{26}\right)+\left(2\times {10}^{26}\right)+\left(4\times {10}^{26}\right)=8\times {10}^{26}$

To find the number of moles $\left(n\right)$, you have to divide the number of atoms by Avogadro’s number.

$$\begin{gathered} \mathrm{n}=\frac{ \mathrm{Number} \mathrm{of} \mathrm{atoms}}{\mathrm{Avogadro}\text{'}\mathrm{s} \mathrm{number}} \\ =\frac{8\times {10}^{26}}{6.02\times {10}^{23}} \\ =1328.90 \mathrm{mol} \mathrm{of} \mathrm{atoms} \end{gathered}$$

To the nearest order of magnitude, it would be 1000 moles.

Thus, the number of moles in a large domestic cat is 1000.