File ended while scanning use of frac ошибка

The main issue is that you mustn’t have blank lines in an equation environment — or any display-math environment, for that matter.

An additional concern: If you use Times Roman as the text font, you should probably also load a Times Roman math font. The mathptmx and newtxtext/newtxmath packages may be suitable.

\documentclass[12pt]{report}
\usepackage[doublespacing,nodisplayskipstretch]{setspace}
\usepackage{newtxtext,newtxmath} % Times Roman text and math fonts
\usepackage[vmargin=3cm, left=4cm, right=2cm]{geometry}
\usepackage{graphicx}
\usepackage{amsmath,amssymb}

% For tighter spacing around \left/\right pairs of parentheses:
\usepackage{mleftright}  \mleftright 

\begin{document}
\begin{equation} \label{eq2}
\frac{\partial N}{\partial t}+\frac{\partial}{\partial x}\left(\frac{MN}{D}\right)
 +\frac{\partial}{\partial y}\left(\frac{N^2}{D}\right)+ gD\frac{\partial z}{\partial y}
 +\frac{\tau_{y}}{p}=0
\end{equation}

\end{document}

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(e\right)}+
    \mathcal{B}_{y}^{\left(e\right)}\right]J\left(J+1\right)+
    \left(\mathcal{B}_{z}^{\left(e\right)}-\frac{1}{2}
    \left[\mathcal{B}_{x}^{\left(e\right)}+\mathcal{B}_{y}^{\left(e\right)}
    \right]\right)\kappa^{2}  \\
E_{k\pm2,k}&=&\frac{1}{4}[\mathcal{B}_{x}^{(e)}-\mathcal{B}_{y}^{(e)}]
    {[J(J+1)-k(k\pm1)]\times[J(J+1)-k(k\pm1)(k\pm2)]}^{\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_{k\pm2,k}&=& \tfrac{1}{4}
    [\mathcal{B}_{x}^{(e)}-\mathcal{B}_{y}^{(e)}]
    [J(J+1)-k(k\pm1)]\times
    \bigl[J(J+1)-k(k\pm1)(k\pm2)\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_{k\pm2,k} &= \tfrac{1}{4}
    [\mathcal{B}_{x}^{(e)}-\mathcal{B}_{y}^{(e)}]
    [J(J+1)-k(k\pm1)]\times
    \bigl[J(J+1)-k(k\pm1)(k\pm2)\bigr]^{1/2}
\end{align}
\end{document}

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} = .739\frac{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}{mol\ETC.} >! File ended while scanning use of \frac.

Latex Error: File ended while scanning use of \frac

Solution:

When marking important points, \frac{}, one missing}.

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