Question

An average cup of coffee contains about $W = 125mg$ caffeine,${C_8}{H_{10}}{N_4}{O_2}$,$6.44 \times {10^{ - x}}$ moles of caffeine are in a cup. Then, the value of x is:

Answer 1

Hint:Moles of caffeine is given in the question and weight of caffeine is also given. So we have to find moles of caffeine again and then equate them.
Moles of caffeine can be calculated using this equation $\dfrac{\text{weight of caffeine( in coffee)}}{\text{molar mass(caffeine)}}$ .

Complete answer:
One mole of a substance is equal to $6.023 \times {10^{23}}$ is known as Avogadro’s number or constant (${N_A}$).
The concept of mole can be used to convert between mass and number of particles.
How do we calculate mole?
We can calculate mole, following methods
I. Number of moles =$\dfrac{\text{no.of atoms or molecules}}{{6.022 \times {{10}^{23}}({N_A})}}$
II. Number of moles= \[\dfrac{\text{ mass of the substance(gm)}}{\text{molar mass of substance}}\]
III. Number of moles=\[\dfrac{\text{ mass of the sample}}{\text{molar mass}}\]
The moles of the caffeine
Moles=\[\dfrac{\text{weight of caffeine}}{\text{molar mass(M)}}\]
Formula of caffeine is given ${C_8}{H_{10}}{N_4}{O_2}$
We can calculate molar mass by this formula
$M = 8 \times 12 + 10 \times 1 + 4 \times 14 + 2 \times 16$
$M = 96 + 10 + 56 + 32$
$M = 192$
$\text{molar mass of C} = 12$
$ H = 1$
$N = 14$
$O = 16$

Additional information: The mole is widely used in chemistry as a convenient way to express amounts of reactants and products of chemical reactions. The term gram-mole was formerly used for “mole of molecules” and gram-atom for “mole of atoms”. The molar fraction or mole fraction of a substance in a mixture is the number of moles of the compound in one sample of the mixture divided by the total number of moles of all components.
Weight of caffeine is given $W = 125mg$
Moles =$\dfrac{{125 \times {{10}^{ - 3}}gm}}{{192}}$
In the question, moles of caffeine are given.
So, both moles must be equal.
$\dfrac{{125 \times {{10}^{ - 3}}}}{{192}} = 6.44 \times {10^{ - x}}$
$6.44 \times {10^{ - 4}} = 6.44 \times {10^{ - x}}$
$x = 4$

Note:
A mole corresponds to the mass of a substance that contains $6.023 \times {10^{23}}$ particles of the substance. The mole is the SI unit for the amount of a substance.