To compare the mass of an electron to that of a proton, we need to find the ratio of the mass of an electron to the mass of a proton.
Here are the given masses:
- Mass of an electron ([tex]\(m_e\)[/tex]): [tex]\(9.1 \times 10^{-31} \, \text{kg}\)[/tex]
- Mass of a proton ([tex]\(m_p\)[/tex]): [tex]\(1.67 \times 10^{-27} \, \text{kg}\)[/tex]
Steps to find the ratio:
1. Identify the masses:
- Electron mass: [tex]\(9.1 \times 10^{-31} \, \text{kg}\)[/tex]
- Proton mass: [tex]\(1.67 \times 10^{-27} \, \text{kg}\)[/tex]
2. Setup the ratio formula:
[tex]\[
\text{Ratio} = \frac{\text{mass of an electron}}{\text{mass of a proton}} = \frac{m_e}{m_p}
\][/tex]
3. Plug in the values:
[tex]\[
\text{Ratio} = \frac{9.1 \times 10^{-31} \, \text{kg}}{1.67 \times 10^{-27} \, \text{kg}}
\][/tex]
4. Simplify the ratio:
- Divide the coefficients: [tex]\(\frac{9.1}{1.67} \approx 0.5449101796407185\)[/tex]
- Divide the exponents: [tex]\(\frac{10^{-31}}{10^{-27}} = 10^{-31 - (-27)} = 10^{-4}\)[/tex]
So we can write:
[tex]\[
\frac{9.1 \times 10^{-31}}{1.67 \times 10^{-27}} = 0.5449101796407185 \times 10^{-4}
\][/tex]
5. Interpret the result:
[tex]\[
0.5449101796407185 \approx 5.4491 \times 10^{-4}
\][/tex]
This simplification shows that the mass of an electron is approximately [tex]\( 0.00054491 \)[/tex] times the mass of a proton. This indicates that the electron is significantly lighter than the proton.
Thus, the electron's mass is about 0.0005449101796407185 times the mass of a proton. These values give a clear comparison of how much smaller the electron's mass is relative to the proton's mass.
