To determine the value of [tex]\( x \)[/tex] in the formula of the amino acid [tex]\( C_nH_xN_xO_2 \)[/tex], where [tex]\( M = 174\, \text{g/mol} \)[/tex], given that it contains [tex]\( 8.1\% \)[/tex] by mass of H and [tex]\( 41.3\% \)[/tex] by mass of [tex]\( C \)[/tex], we can follow these steps:
1. Calculate the mass of hydrogen (H) in one mole of the amino acid:
Given that hydrogen makes up [tex]\( 8.1\% \)[/tex] of the total mass,
[tex]\[
\text{mass of H} = 0.081 \times 174\, \text{g} = 14.094\, \text{g}
\][/tex]
2. Calculate the number of moles of hydrogen atoms:
The atomic mass of hydrogen (H) is [tex]\( 1\, \text{g/mol} \)[/tex],
[tex]\[
\text{moles of H} = \frac{14.094\, \text{g}}{1\, \text{g/mol}} = 14.094\, \text{mol}
\][/tex]
3. Calculate the mass of carbon (C) in one mole of the amino acid:
Given that carbon makes up [tex]\( 41.3\% \)[/tex] of the total mass,
[tex]\[
\text{mass of C} = 0.413 \times 174\, \text{g} = 71.862\, \text{g}
\][/tex]
4. Calculate the number of moles of carbon atoms:
The atomic mass of carbon (C) is [tex]\( 12\, \text{g/mol} \)[/tex],
[tex]\[
\text{moles of C} = \frac{71.862\, \text{g}}{12\, \text{g/mol}} = 5.9885\, \text{mol}
\][/tex]
5. Determine the remainder of the mass for N and O:
Since the amino acid contains [tex]\( C_nH_xN_xO_2 \)[/tex], the number of oxygen atoms is fixed at 2.
The atomic mass of oxygen (O) is [tex]\( 16\, \text{g/mol} \)[/tex],
[tex]\[
\text{mass of O}_2 = 2 \times 16\, \text{g/mol} = 32\, \text{g}
\][/tex]
Subtract the mass of C, H, and O from the total mass to find the mass of nitrogen (N):
[tex]\[
\text{mass of NxO}_2 = 174\, \text{g} - (71.862\, \text{g} + 14.094\, \text{g}) = 88.044\, \text{g}
\][/tex]
[tex]\[
\text{mass of N} = 88.044\, \text{g} - 32\, \text{g} = 56.044\, \text{g}
\][/tex]
6. Calculate the number of moles of nitrogen atoms:
The atomic mass of nitrogen (N) is [tex]\( 14\, \text{g/mol} \)[/tex],
[tex]\[
\text{moles of N} = \frac{56.044\, \text{g}}{14\, \text{g/mol}} = 4.0031\, \text{mol}
\][/tex]
7. Determine the value of [tex]\( x \)[/tex]:
In the given molecular formula [tex]\( C_nH_xN_xO_2 \)[/tex], the number of hydrogen atoms [tex]\( x \)[/tex] is already given by the moles of hydrogen atoms calculated previously,
[tex]\[
x = 14.094
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] in the formula is approximately [tex]\( 14.1 \)[/tex] (rounded to one decimal place).
