Q42P - Fundamentals Of Physics

Question

One molecule of water${}^{\left(H_{2}O\right)}$ contains two atoms of hydrogen and one atom of oxygen. A hydrogen atom has a mass of${}^{\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{u}}$ and an atom of oxygen has a mass of ${}^{\mathbf{16}\mathbf{u}}$ , approximately. (a) What is the mass in kilograms of one molecule of water? (b) How many molecules of water are in the world’s oceans, which have an estimated total mass of ${}^{1.4\times {10}^{21}\mathbf{ }\mathbf{kg}}$ ?

Step-by-Step Solution

Answer

  ( a ) The mass of one molecule of water is ${}^{\mathbf{30}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{27}}\mathbf{ }\mathbf{kg}}$ .

  ( b )  Number of molecules of water present in the world’s oceans is ${}^{\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{46}}\mathbf{ }\mathbf{kg}}$ .

Step 1: Given data

The mass of the hydrogen atom is ${}^{\mathbf{1}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{u}}$ .

The mass of the oxygen atom is ${}^{\mathbf{16}\mathbf{ }\mathbf{u}}$ .

The mass of water in the ocean is ${}^{\mathbf{1}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{21}}\mathbf{ }\mathbf{kg}}$ .

Step 2: Understanding the unit conversion

Unit conversion is the conversion between different units of measure for the same quantity, usually through multiplier conversion factors.

Step 3: (a) Determination of the mass in kg.

The atomic mass unit of H₂O is,

$\mathbf{16}\mathbf{ }\mathbf{u}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{1}\mathbf{ }\mathbf{u}\mathbf{ }\mathbf{+}\mathbf{ }\mathbf{1}\mathbf{ }\mathbf{u}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{18}\mathbf{ }\mathbf{u}$

Here, ${}^{\mathbf{16}\mathbf{ }\mathbf{u}}$ is the atomic mass unit of oxygen and ${}^{\mathbf{1}\mathbf{ }\mathbf{u}}$ is the atomic mass unit of hydrogen. 

It is known that, ${}^{\mathbf{1}\mathbf{ }\mathbf{u}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{1}\mathbf{.}\mathbf{66053886}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{27}}}$

Now, convert it into kilograms.

Mass of water molecule m in kg is calculated as,

$$\begin{gathered} \mathbf{m}\mathbf{ }\mathbf{=}\left(18 u \times \frac{1.66053886\times {10}^{-27} \mathrm{kg}}{1 u}\right) \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{29}\mathbf{.}\mathbf{889}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{27}}\mathbf{ }\mathbf{kg} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\approx }\mathbf{30}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{27}}\mathbf{ }\mathbf{kg} \end{gathered}$$

Thus, the mass of the water molecule is${}^{\mathbf{30}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{27}}}$.

Step 4: (b) Determination of the number of molecules.

For the calculation of the number of molecules of water present in the world’s oceans with some estimated mass, divide the total mass by the mass of each molecule and obtain the (approximate) number of water molecules. Therefore,

$$\begin{gathered} \mathbf{Number}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{molecules}\mathbf{ }\mathbf{=}\frac{\mathbf{Totel}\mathbf{ }\mathbf{mass}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{ocean}}{\mathbf{Mass}\mathbf{ }\mathbf{of}\mathbf{ }\mathbf{one}\mathbf{ }\mathbf{water}\mathbf{ }\mathbf{molecule}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\frac{\mathbf{1}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{21}}\mathbf{ }\mathbf{kg}}{\mathbf{30}\mathbf{.}\mathbf{0}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{27}}\mathbf{ }\mathbf{kg}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{0466}\mathbf{\times }{\mathbf{10}}^{\mathbf{48}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\approx }\mathbf{4}\mathbf{.}\mathbf{67}\mathbf{\times }{\mathbf{10}}^{\mathbf{46}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{\approx }\mathbf{ }\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{46}} \end{gathered}$$

Thus, the number of molecules is ${}^{\mathbf{5}\mathbf{\times }{\mathbf{10}}^{\mathbf{46}}}$ .