Let's complete each of the given nuclear reactions step-by-step using the conservation of mass number (A) and atomic number (Z).
### Part (a):
Given the reaction:
[tex]\[ {}_{13}^{27} \text{Al} + {}_{2}^{4} \text{He} \longrightarrow ? + {}_{0}^{1} \text{n} \][/tex]
1. Conservation of mass number (A):
[tex]\[ 27 + 4 = A(?) + 1 \][/tex]
[tex]\[ 31 = A(?) + 1 \][/tex]
[tex]\[ A(?) = 30 \][/tex]
2. Conservation of atomic number (Z):
[tex]\[ 13 + 2 = Z(?) + 0 \][/tex]
[tex]\[ 15 = Z(?) \][/tex]
Thus, the unknown element is:
[tex]\[ {}_{15}^{30}\text{P} \][/tex]
So, the completed equation is:
[tex]\[ {}_{13}^{27} \text{Al} + {}_{2}^{4} \text{He} \longrightarrow {}_{15}^{30}\text{P} + {}_{0}^{1} \text{n} \][/tex]
### Part (b):
Given the reaction:
[tex]\[ {}_{04}^{239} \text{P1I} + ? \longrightarrow {}_{36}^{242} \text{Cm} + \frac{1}{6} \text{m} \][/tex]
1. Conservation of mass number (A):
[tex]\[ 239 + A(?) = 242 + 6 \][/tex]
[tex]\[ 239 + A(?) = 248 \][/tex]
[tex]\[ A(?) = 9 \][/tex]
2. Conservation of atomic number (Z):
[tex]\[ 4 + Z(?) = 36 \][/tex]
[tex]\[ Z(?) = 32 \][/tex]
Thus, the unknown element is:
[tex]\[ {}_{32}^{9}\text{Ge} \][/tex]
So, the completed equation is:
[tex]\[ {}_{04}^{239} \text{P1I} + {}_{32}^{9}\text{Ge} \longrightarrow {}_{36}^{242} \text{Cm} + \frac{1}{6} \text{m} \][/tex]
### Part (c):
Given the reaction:
[tex]\[ {}_{7}^{14} \text{N} + {}_{2}^{4} \text{He} \longrightarrow ? + {}_{1}^{1} \text{H} \][/tex]
1. Conservation of mass number (A):
[tex]\[ 14 + 4 = A(?) + 1 \][/tex]
[tex]\[ 18 = A(?) + 1 \][/tex]
[tex]\[ A(?) = 17 \][/tex]
2. Conservation of atomic number (Z):
[tex]\[ 7 + 2 = Z(?) + 1 \][/tex]
[tex]\[ 9 = Z(?) + 1 \][/tex]
[tex]\[ Z(?) = 8 \][/tex]
Thus, the unknown element is:
[tex]\[ {}_{8}^{17}\text{O} \][/tex]
So, the completed equation is:
[tex]\[ {}_{7}^{14} \text{N} + {}_{2}^{4} \text{He} \longrightarrow {}_{8}^{17}\text{O} + {}_{1}^{1} \text{H} \][/tex]
### Part (d):
Given the reaction:
[tex]\[ {}_{92}^{235} \text{U} \longrightarrow ? + {}_{55}^{135} \text{Cs} + 4{}_{0}^{1} \text{n} \][/tex]
1. Conservation of mass number (A):
[tex]\[ 235 = A(?) + 135 + 4 \][/tex]
[tex]\[ 235 = A(?) + 139 \][/tex]
[tex]\[ A(?) = 96 \][/tex]
2. Conservation of atomic number (Z):
[tex]\[ 92 = Z(?) + 55 \][/tex]
[tex]\[ Z(?) = 37 \][/tex]
Thus, the unknown element is:
[tex]\[ {}_{37}^{96}\text{Rb} \][/tex]
So, the completed equation is:
[tex]\[ {}_{92}^{235} \text{U} \longrightarrow {}_{37}^{96}\text{Rb} + {}_{55}^{135} \text{Cs} + 4{}_{0}^{1} \text{n} \][/tex]
In summary, the completed equations are:
a. [tex]\[ {}_{13}^{27} \text{Al} + {}_{2}^{4} \text{He} \longrightarrow {}_{15}^{30} \text{P} + {}_{0}^{1} \text{n} \][/tex]
b. [tex]\[ {}_{04}^{239} \text{P1I} + {}_{32}^{9} \text{Ge} \longrightarrow {}_{36}^{242} \text{Cm} + \frac{1}{6} \text{m} \][/tex]
c. [tex]\[ {}_{7}^{14} \text{N} + {}_{2}^{4} \text{He} \longrightarrow {}_{8}^{17} \text{O} + {}_{1}^{1} \text{H} \][/tex]
d. [tex]\[ {}_{92}^{235} \text{U} \longrightarrow {}_{37}^{96} \text{Rb} + {}_{55}^{135} \text{Cs} + 4{}_{0}^{1} \text{n} \][/tex]
