To determine how many molecules are in a 23.45 g sample of copper (II) hydroxide, [tex]\( \text{Cu(OH)}_2 \)[/tex], we need to follow a series of steps involving dimensional analysis. Here’s how we break it down:
1. Determine the number of moles of [tex]\( \text{Cu(OH)}_2 \)[/tex] in the sample:
Given:
- Mass of the sample ([tex]\( m \)[/tex]): 23.45 g
- Molar mass of [tex]\( \text{Cu(OH)}_2 \)[/tex] ([tex]\( M \)[/tex]): 97.562 g/mol
The formula to find the number of moles ([tex]\( n \)[/tex]) is:
[tex]\[
n = \frac{m}{M}
\][/tex]
Substituting the given values:
[tex]\[
n = \frac{23.45 \text{ g}}{97.562 \text{ g/mol}} \approx 0.24035997622024968 \text{ mol}
\][/tex]
2. Calculate the number of molecules using Avogadro’s number:
Avogadro’s number ([tex]\( N_A \)[/tex]) tells us that one mole of any substance contains [tex]\( 6.022 \times 10^{23} \)[/tex] molecules.
The number of molecules ([tex]\( N \)[/tex]) can be calculated using the formula:
[tex]\[
N = n \times N_A
\][/tex]
Substituting the previously calculated number of moles and Avogadro’s number:
[tex]\[
N = 0.24035997622024968 \text{ mol} \times 6.022 \times 10^{23} \text{ molecules/mol}
\][/tex]
Performing the multiplication:
[tex]\[
N \approx 1.4474477767983435 \times 10^{23} \text{ molecules}
\][/tex]
3. Expressing the answer with the correct number of significant figures:
The given mass of the sample (23.45 g) has 4 significant figures, and Avogadro's number is generally given to 4 significant figures for simplicity.
Therefore, the number of molecules should also be expressed with 4 significant figures:
[tex]\[
N \approx 1.447 \times 10^{23} \text{ molecules}
\][/tex]
So, in a 23.45 g sample of copper (II) hydroxide ([tex]\( \text{Cu(OH)}_2 \)[/tex]), there are approximately [tex]\( 1.447 \times 10^{23} \)[/tex] molecules.
