File ended while scanning use of frac ошибка

I am unable to generate the error message you say you’re getting. I would like to suggest, though, that you rewrite some of the code to use left and right much more sparingly than you do at present. If you compare the LaTeX codes for the first and second groups of equations, you will hopefully agree that the second group’s code is much simpler and easier to read.

A separate comment: The IEEEeqnarray environment is extremely powerful and flexible. However, at present you don’t really seem to be making use of the package’s machinery. You may therefore wish to look into making use of the simpler align environment, which is provided by the amsmath package. As the second and third groups of equations in the following screenshot demonstrate, the output is identical — though, arguably, the syntax of the align environment is simpler.

documentclass{article}  
usepackage{IEEEtrantools} % for 'IEEEeqnarray' env.
usepackage{geometry,amsmath}
begin{document}

OP's original form
begin{IEEEeqnarray}{rCl}label{key1}
E_{k,k}&=&frac{1}{2}left[mathcal{B}_{x}^{left(eright)}+
    mathcal{B}_{y}^{left(eright)}right]Jleft(J+1right)+
    left(mathcal{B}_{z}^{left(eright)}-frac{1}{2}
    left[mathcal{B}_{x}^{left(eright)}+mathcal{B}_{y}^{left(eright)}
    right]right)kappa^{2}  \
E_{kpm2,k}&=&frac{1}{4}[mathcal{B}_{x}^{(e)}-mathcal{B}_{y}^{(e)}]
    {[J(J+1)-k(kpm1)]times[J(J+1)-k(kpm1)(kpm2)]}^{frac{1}{2}}
end{IEEEeqnarray}

bigskip
Suggested edits
begin{IEEEeqnarray}{rCl}label{key2}
E_{k,k}&=& tfrac{1}{2}
    [mathcal{B}_{x}^{(e)}+mathcal{B}_{y}^{(e)}]J(J+1)
    +bigl( mathcal{B}_{z}^{(e)} -tfrac{1}{2}
    [mathcal{B}_{x}^{(e)}+mathcal{B}_{y}^{(e)}]
    bigr)kappa^{2}  \
E_{kpm2,k}&=& tfrac{1}{4}
    [mathcal{B}_{x}^{(e)}-mathcal{B}_{y}^{(e)}]
    [J(J+1)-k(kpm1)]times
    bigl[J(J+1)-k(kpm1)(kpm2)bigr]^{1/2}
end{IEEEeqnarray}

bigskip
Solution that uses an texttt{align} environment
begin{align}label{key3}
E_{k,k} &= tfrac{1}{2}
    [mathcal{B}_{x}^{(e)}+mathcal{B}_{y}^{(e)}]J(J+1)
    +bigl( mathcal{B}_{z}^{(e)} -tfrac{1}{2}
    [mathcal{B}_{x}^{(e)}+mathcal{B}_{y}^{(e)}]
    bigr)kappa^{2}  \
E_{kpm2,k} &= tfrac{1}{4}
    [mathcal{B}_{x}^{(e)}-mathcal{B}_{y}^{(e)}]
    [J(J+1)-k(kpm1)]times
    bigl[J(J+1)-k(kpm1)(kpm2)bigr]^{1/2}
end{align}
end{document}

Related videos on Youtube

Related

I started using laTeX today. Can I get some advice on how to fix these compiling errors in TeXstudio?

I keep getting the compiling error >file ended while scanning use of frac. I can’t see any missing {}, so I’m not sure what to do.

documentclass[]{scrreprt}
usepackage{geometry}
begin{document}

Molar Mass of Nitrogen
begin{math}
\ Bulb Volume: 213.7 cm^{3} = .2317 L
\ Internal bulb pressure: 20 inHg = 508 mmHg = .508 bar
\ Temperature: 294.2 K
\ M = frac{rho}{P}RT
\ rho = frac{.158 g}{.2137 L} = .739frac{g}{L}
\ M = frac{.739 frac{g}{L}}{.508 bar} * .083144 frac{L*bar}{mol*K} * 294.2 K = 35.6 frac{g}{mol}

\ Calculation of Volume by van der Waals Equation with Successive Approximations
\ V = frac{nRT}{P + frac{n^{2}a}{V^{2}}} + nb
\ a = 1.408 frac{L^{2}*bar}{mol^{2}}
b = .03913 L
\ V approx frac{nRT}{P} + nb approx frac{nRT}{P}
\ n = frac{m}{M} = frac{.158 g}{21.35 frac{g}{mol}} = 7.4*10^{-3} mol
\ V = frac{7.4*10^{-3} mol * .083144 frac{L*bar}{mol*K} * 294.2 K}{.508 bar} = .2138 L
\ V = frac{7.4*10^{-3} mol * .083144 frac{L*bar}{mol*K} * 294.2 K}{.508 bar + frac{(7.4*10^{-3} mol)^{2}*1.408 frac{L^{2}*bar}{mol^{2}}} {(.2138 L)^{2}} + 7.4*10^{-3} mol * .03913 L = .2137 L
end{math}
end{document}          

EDIT: This is for a calculations addendum to an undergraduate chemistry lab. I figured it would be worth the time investment to learn laTeX. It’s already significantly faster to do calculations in it than word, if only I can get around these compiling errors.

Relevant log text:

Runaway argument? {.508 bar + frac {(7.4*10^{-3} mol)^{2}*1.408 frac {L^{{2}*bar}{molETC.} >! File ended while scanning use of frac.