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To determine the sum of the coefficients when the chemical equation [tex]\( FeCr_2O_4 + K_2CO_3 + O_2 \rightarrow K_2CrO_4 + Fe_2O_3 + CO_2 \)[/tex] is balanced with the smallest set of integers, let me guide you through the step-by-step process.
1. Write the unbalanced equation:
[tex]\[ FeCr_2O_4 + K_2CO_3 + O_2 \rightarrow K_2CrO_4 + Fe_2O_3 + CO_2 \][/tex]
2. Balance the elements one by one.
- Balance Chromium (Cr):
There are 2 Cr atoms in [tex]\( FeCr_2O_4 \)[/tex]. Thus, we need 2 [tex]\( K_2CrO_4 \)[/tex] on the products side to balance Chromium.
[tex]\[ FeCr_2O_4 + K_2CO_3 + O_2 \rightarrow 2K_2CrO_4 + Fe_2O_3 + CO_2 \][/tex]
- Balance Iron (Fe):
There is 1 Fe in [tex]\( FeCr_2O_4 \)[/tex], so we need to balance the number of Fe atoms on both sides, thus 1 [tex]\( Fe_2O_3 \)[/tex] is needed:
[tex]\[ 2FeCr_2O_4 + K_2CO_3 + O_2 \rightarrow 2K_2CrO_4 + 1Fe_2O_3 + CO_2 \][/tex]
- Balance Potassium (K):
Since each [tex]\( K_2CO_3 \)[/tex] contains 2 K atoms, and we need a total of 4 K atoms (from 2 [tex]\( K_2CrO_4 \)[/tex]), thus 4 [tex]\( K_2CO_3 \)[/tex] are needed:
[tex]\[ 2FeCr_2O_4 + 4K_2CO_3 + O_2 \rightarrow 2K_2CrO_4 + 1Fe_2O_3 + 4CO_2 \][/tex]
- Balance Oxygen (O):
The total number of oxygen atoms on the reactants side (counting from each compound) must equal the total on the products side:
[tex]\[ 2FeCr_2O_4 (8) + 4K_2CO_3 (12) + O_2 (14) \rightarrow 2K_2CrO_4 (10) + 1Fe_2O_3 (3) + 4CO_2 (8) \][/tex]
We see that the count balances out when 7 [tex]\( O_2 \)[/tex] are added:
[tex]\[ 2FeCr_2O_4 + 4K_2CO_3 + 7O_2 \rightarrow 2K_2CrO_4 + Fe_2O_3 + 4CO_2 \][/tex]
3. Sum the coefficients of the balanced equation:
[tex]\[ 2FeCr_2O_4 + 4K_2CO_3 + 7O_2 \rightarrow 2K_2CrO_4 + 1Fe_2O_3 + 4CO_2 \][/tex]
The coefficients are [tex]\( 2, 4, 7, 2, 1, 4 \)[/tex].
4. Add these coefficients together:
[tex]\[ 2 + 4 + 7 + 2 + 1 + 4 = 19 \][/tex]
So, the correct answer when the equation is balanced, the sum of the coefficients is 19.
1. Write the unbalanced equation:
[tex]\[ FeCr_2O_4 + K_2CO_3 + O_2 \rightarrow K_2CrO_4 + Fe_2O_3 + CO_2 \][/tex]
2. Balance the elements one by one.
- Balance Chromium (Cr):
There are 2 Cr atoms in [tex]\( FeCr_2O_4 \)[/tex]. Thus, we need 2 [tex]\( K_2CrO_4 \)[/tex] on the products side to balance Chromium.
[tex]\[ FeCr_2O_4 + K_2CO_3 + O_2 \rightarrow 2K_2CrO_4 + Fe_2O_3 + CO_2 \][/tex]
- Balance Iron (Fe):
There is 1 Fe in [tex]\( FeCr_2O_4 \)[/tex], so we need to balance the number of Fe atoms on both sides, thus 1 [tex]\( Fe_2O_3 \)[/tex] is needed:
[tex]\[ 2FeCr_2O_4 + K_2CO_3 + O_2 \rightarrow 2K_2CrO_4 + 1Fe_2O_3 + CO_2 \][/tex]
- Balance Potassium (K):
Since each [tex]\( K_2CO_3 \)[/tex] contains 2 K atoms, and we need a total of 4 K atoms (from 2 [tex]\( K_2CrO_4 \)[/tex]), thus 4 [tex]\( K_2CO_3 \)[/tex] are needed:
[tex]\[ 2FeCr_2O_4 + 4K_2CO_3 + O_2 \rightarrow 2K_2CrO_4 + 1Fe_2O_3 + 4CO_2 \][/tex]
- Balance Oxygen (O):
The total number of oxygen atoms on the reactants side (counting from each compound) must equal the total on the products side:
[tex]\[ 2FeCr_2O_4 (8) + 4K_2CO_3 (12) + O_2 (14) \rightarrow 2K_2CrO_4 (10) + 1Fe_2O_3 (3) + 4CO_2 (8) \][/tex]
We see that the count balances out when 7 [tex]\( O_2 \)[/tex] are added:
[tex]\[ 2FeCr_2O_4 + 4K_2CO_3 + 7O_2 \rightarrow 2K_2CrO_4 + Fe_2O_3 + 4CO_2 \][/tex]
3. Sum the coefficients of the balanced equation:
[tex]\[ 2FeCr_2O_4 + 4K_2CO_3 + 7O_2 \rightarrow 2K_2CrO_4 + 1Fe_2O_3 + 4CO_2 \][/tex]
The coefficients are [tex]\( 2, 4, 7, 2, 1, 4 \)[/tex].
4. Add these coefficients together:
[tex]\[ 2 + 4 + 7 + 2 + 1 + 4 = 19 \][/tex]
So, the correct answer when the equation is balanced, the sum of the coefficients is 19.
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