Mfcttrf

Consider the following MWE

documentclass[border=10 pt]standalone
usepackagemathtools,setspace

begindocument
$displaystyle
setstretch1.5
beginarrayccccccccccccccccccccccc 
^12 mathrmC&+&^1 mathrmH &longrightarrow& ^13mathrmN&+&gamma\
&&&&^13 mathrmN& longrightarrow&^13 mathrmC&+&e^+&+&nu\
&&&&&&^13 mathrmC&+&^mathrml mathrmH& longrightarrow&^14 mathrmN&+&gamma\
&&&&&&&&&&^14 mathrmN&+&^mathrml mathrmH& longrightarrow&^15 mathrmO&+&gamma\
&&&&&&&&&&&&&&^15 mathrmO& longrightarrow&^15 mathrmN&+&e^+&+&nu\
&&&&&&&&&&&&&&&&^15 mathrmN&+&^1 mathrmH &longrightarrow&^12 mathrmC&+&^4 mathrmHe
endarray
$ 
enddocument

with the output

The output is pretty much what I'm looking for, but the way I get it seems rather clunky to me... Is there a better way to get the same output? The only relevant part here is that the same elements in each reaction need to be below each other... The spacing of the other parts doesn't have to be the same as in the picture..

It's a matter of opinion whether this is less clunky...it eliminates all & tab characters, but requires multiple stackunders and some bracing.

The appearance is a bit different, however "the same elements of each reaction are below each other", as requested by the OP. While there are no other alignments that apply one may observe that the spacing on any given line is "correct".

documentclass[border=10 pt]standalone
usepackagemathtools,setspace,stackengine
begindocument
renewcommandstackalignmentl
renewcommandstacktypeL
setstackgapL1.4baselineskip
stackMath
$displaystyle
^12mathrmC+^1mathrmHlongrightarrow stackunder^13mathrmN+gamma
 ^13mathrmNlongrightarrow stackunder^13mathrmC+e^++nu
 ^13mathrmC+^mathrmlmathrmHlongrightarrow stackunder^14mathrmN+gamma
 ^14mathrmN+^mathrmlmathrmHlongrightarrow stackunder^15mathrmO+gamma
 ^15mathrmOlongrightarrow stackunder^15mathrmN+e^++nu
 ^15mathrmN+^1mathrmHlongrightarrow ^12mathrmC+^4mathrmHe

$
enddocument

I suggest to align your entries like the way I did (using Tab) to make it easier to read and modify if needed.
I have used newcommand to create a way for you to enter the data of a row and every cell in the array is encapsulated by mathrm....
Also, I changed the 23 cs you had in the array to *23c.

documentclass[border=10 pt]standalone
usepackagemathtools,setspace
% Usage:
% dataoperand-1operator-1operand-2operator-2operand-3operator-3operand-4
% Simple example:
% dataa+b-c*d
newcommanddata[7]%
 ensuremath%
 mathrm#1 & mathrm#2 & mathrm#3 & mathrm#4 & mathrm#5 & mathrm#6 & mathrm#7
 %
%
begindocument
$displaystyle
setstretch1.5
beginarray*23c 
 data^12 C+ ^1 H longrightarrow ^13 N +gamma \
&&&& data^13 Nlongrightarrow ^13 C + e^+ +nu \
&&&&&& data^13 C+ ^1 H longrightarrow ^14 N +gamma \
&&&&&&&&&& data^14 N+ ^1 H longrightarrow ^15 O +gamma \
&&&&&&&&&&&&&& data^15 Olongrightarrow ^15 N + e^+ +nu \
&&&&&&&&&&&&&&&& data^15 N+ ^1 H longrightarrow ^12 C +^4 He\
endarray
$
enddocument